Recent Developments on the First Chen Inequality in Differential Geometry

Abstract

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first $\delta$-invariant, $\delta \left(2\right)$, and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades.

1. Introduction

In 1936, H. Whitney [1] proved that every n-dimensional (differentiable) manifold can be immersed in the Euclidean $2n$-space ${\mathbb{E}}^{2n}$, and embedded it in ${\mathbb{E}}^{2n+1}$ as a closed set.

One of the most fundamental problems in submanifold theory is the problem of isometric immersibility. The earliest publication on isometric embedding appeared in 1873 by L. Schläfli [2] in which he asserted that any Riemannian n-manifold can be isometrically embedded in the Euclidean space of dimension $\frac{1}{2}n(n+1)$. Apparently, it is appropriate to assume that Schläfli had in mind analytic metrics and local analytic embeddings. This was later known as the Schläfli conjecture.

M. Janet [3] published, in 1926, a proof of Schläfli’s conjecture, which states that every real analytic Riemannian n-manifold M can be locally isometrically embedded into any real analytic Riemannian manifold of the dimension $\frac{1}{2}n(n+1)$. É. Cartan [4] revised, in 1927, Janet’s paper with the same title; while Janet wrote the problem in the form of a system of partial differential equations, which he investigated using rather complicated methods, Cartan applied his own theory of Pfaffian systems in involution. Both Janet’s and Cartan’s proofs contained obscurities. In 1931, C. Burstin [5] got rid of them. Cartan–Janet’s result implies that every Einstein n-manifold with $n\ge 3$ can be locally isometrically embedded in ${\mathbb{E}}^{n(n+1)/2}$ since Einstein metrics are real analytic (see, e.g., [6]).

Cartan–Janet’s result is, dimensionwise, the best possible so that there are n-dimensional real analytic Riemannian manifolds that do not admit local isometric embeddings in any Euclidean space of dimension less than $\frac{1}{2}n(n+1)$. Further, not every Riemannian n-manifold can be isometrically immersed in ${\mathbb{E}}^{\frac{1}{2}n(n+1)}$. For instance, not every Riemannian two-manifold can be isometrically immersed in ${\mathbb{E}}^3$.

In 1956, J. F. Nash made the following major contribution to the theory of submanifolds.

Theorem 1

([7]). Every Riemannian n-manifold can be isometrically embedded in a Euclidean m space with the dimension $m=\frac{n}{2}(n+1)(3n+11)$.

The aim of this embedding theorem was the hope that if every Riemannian manifold could be regarded as a Euclidean submanifold, then it could yield the opportunity to use help from extrinsic geometry. But this hope was not materialized for many years as pointed out, in 1985, by M. Gromov’s article [8]. There were several reasons why it is so difficult to apply Nash’s embedding theorem. One reason is that it requires, in general, a very large codimension for a Riemannian manifold to admit an isometric embedding in a Euclidean space. On the other hand, submanifolds of higher codimensions are difficult to understand. The other reason is that, at that time, there existed no general optimal relations between known intrinsic invariants and main extrinsic invariants for arbitrary Euclidean submanifolds (see [9]). Such difficulties led to the following fundamental problem in submanifold theory (see [10,11]).

Problem 1.

Find simple relationships between main extrinsic invariants and main intrinsic invariants of a submanifold.

To provide a solution to this problem, in 1993, the first author introduced a new type of Riemannian invariant, $\delta \left(2\right)$, defined by $\delta \left(2\right)=\tau -infK$ (see [12,13]), where $\tau$ is the scalar curvature and K is the sectional curvature. Applying this invariant, the first author was able to establish the following solution to Problem 1:

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n-2}{2}\left\{\frac{n^2}{n-1}{\parallel H\parallel }^2+(n+1)c\right\}\end{array}$$

(1)

for any submanifold M with $n=dimM\ge 3$ in a real space form $R^m\left(c\right)$ of a constant sectional curvature c, where ${\parallel H\parallel }^2$ is the squared mean curvature of M.

Inequality (1) was later known as the first Chen inequality or Chen’s first inequality. After [12], inequality (1) drew the attention of many geometers. Consequently, many inequalities of a similar type were established by many geometers for different kinds of submanifolds in various ambient manifolds.

In [10,11], the first author established another solution to Problem 1 for n-dimensional submanifolds in a real space form $R^m\left(c\right)$, namely

$$\begin{array}{c}\hfill \mathrm{Ric}\left(X\right)\le \frac{n^2}{4}{\parallel H\parallel }^2+(n-1)c,n\ge 2,\end{array}$$

(2)

where $\mathrm{Ric}\left(X\right)$ is the Ricci curvature at the unit vector $X\in TM$. This inequality was later known as the Chen–Ricci’s inequality.

Recently, the first author and A. M. Blaga [14] provided a comprehensive survey on the recent developments of Chen–Ricci’s inequality. The main purpose of this article is, thus, to present a comprehensive survey of the recent developments of the first Chen inequality. It is both authors’ intention that this survey would provide a useful reference for graduate students, as well as for researchers working on this research topic.

2. Preliminaries

In the following, we denote by $\mathfrak{X}\left(M\right)$ the space of all vector fields on a manifold M, by $\mathcal{F}\left(M\right)$ the space of functions and by $T^1\left(M\right)$ the unit tangent bundle of M. For basic notions and results on Riemannian manifolds and submanifolds, we refer to the books [15,16,17,18,19,20,21].

2.1. Basics on Riemannian Manifolds

Let ∇ be the Levi–Civita connection of an n-dimensional Riemannian manifold $(M,g)$. Then, the Riemann curvature tensor R of M is the tensor field given by

$$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$$

for $X,Y,Z\in \mathfrak{X}\left(M\right)$, where $[,]$ denotes the Lie bracket. For an orthonormal basis $\{X,Y\}$ of a plane section $\pi \subset T_{p}M$, the sectional curvature $K\left(\pi \right)$ is defined by

$$K\left(\pi \right)=g\left(R\right(X,Y)Y,X).$$

The Ricci tensor at point $p\in M$, denoted by ${\mathrm{Ric}}_p$, is given by

$${\mathrm{Ric}}_p(Y,Z)=\mathrm{Trace}\{X\mapsto R(X,Y)Z\}$$

or

$$\mathrm{Ric}(X,Y)=\sum _{i=1}^n〈R(E_i,X)Y,E_i〉,$$

where $\{E_1,\cdots ,E_n\}$ is an orthonormal frame of $TM$. $(M,g)$ is said to be an Einstein manifold if we have $\mathrm{Ric}=cg$ for some constant c. Given a unit vector $X\in TM$, the Ricci curvature $\mathrm{Ric}\left(X\right)$ at X is given by $\mathrm{Ric}\left(X\right)=\mathrm{Ric}(X,X).$

The scalar curvature $\tau$ of M is given by

$$\tau =\sum _{i<j}K(E_i\wedge E_j).$$

2.2. Basics on Riemannian Submanifolds

If M is a submanifold of a Riemannian manifold $(\tilde{M},\tilde{g})$, we denote the Levi–Civita connections of M and $\tilde{M}$ by ∇ and $\tilde{\nabla }$, respectively. The formulas of Gauss and Weingarten of M in $\tilde{M}$ are expressed by

$$\begin{array}{cc}& {\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),{\tilde{\nabla }}_X\xi =-A_{\xi }X+D_X,\xi ,\hfill \end{array}$$

respectively, where $X,Y$ are tangent vector fields and $\xi$ is a normal vector field of M. We call $h,A,D$ the second fundamental form, shape operator and normal connection of M, respectively.

For a normal vector $\xi$ at $p\in M$, the shape operator $A_{\xi }$ is a self-adjoint endomorphism of $T_{p}M$. It is known that the shape operator and the second fundamental form are related by $〈h(X,Y),\xi 〉=〈A_{\xi }X,Y〉,$ where $〈,〉$ is the inner product associated with the metric.

For a submanifold M in $\tilde{M}$, the mean curvature vector is defined by

$$H=\frac{1}{n}\mathrm{Trace}h,n=dimM.$$

The submanifold M is called minimal (resp., totally geodesic) if its mean curvature vector (resp., its second fundamental form) vanishes identically.

Denote the Riemann curvature tensors of M and $\tilde{M}$ by R and $\tilde{R}$, respectively. If $\tilde{M}$ is of constant curvature c, then Gauss, Codazzi and Ricci’s equations are given, respectively, by

$$\begin{array}{c}〈R(X,Y)Z,W〉=\langle A_{h(Y,Z)}X,W\rangle -\langle A_{h(X,Z)}Y,W\rangle \hfill \\ +c(〈X,W〉〈Y,Z〉-〈X,Z〉〈Y,W〉),\hfill \\ \left({\overline{\nabla }}_{X}h\right)(Y,Z)=\left({\overline{\nabla }}_{Y}h\right)(X,Z),\hfill \\ \langle R^{\perp }(X,Y)\xi ,\eta \rangle =〈[A_{\xi },A_{\eta }]X,Y〉\hfill \end{array}$$

for tangent vectors $X,Y,Z,W$ and normal vectors $\xi ,\eta$ of M, where $R^{\perp }$ is the normal curvature tensor and $\overline{\nabla }h$ is defined by $\left({\overline{\nabla }}_{X}h\right)(Y,Z)=D_{X}h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z).$

2.3. Submanifolds of Kähler Manifolds

There are some important types of submanifolds of an almost Hermitian manifold $(\tilde{M},J,g)$; namely, complex, totally real and slant submanifolds based on the action of the almost complex structure J of $\tilde{M}$ acting on the tangent bundle $TM$ of M. More precisely, a submanifold M is called complex (or holomorphic or invariant) if $J\left(T_{p}M\right)=T_{p}M$, $\forall p\in M$; M is called totally real if $J\left(T_{p}M\right)\subseteq T_p^{\perp }M$, $\forall p\in M$ (see [22]). In particular, a totally real submanifold M is called Lagrangian if it satisfies ${dim}_{\mathbb{R}}M={dim}_{\mathbb{C}}\tilde{M}$ (see [23] for general results on Lagrangian submanifolds). In addition, a submanifold M of $(\tilde{M},J,g)$ is called a CR-submanifold (cf. [24]) if there exists a totally real distribution ${\mathcal{D}}^{\perp }$ on M whose orthogonal complement distribution $\mathcal{D}$ is a complex distribution, i.e., $TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp }$ and $J{\mathcal{D}}_p^{\perp }\subset T_p^{\perp }M,\forall p\in M$. A CR-submanifold M is called an antiholomorphic submanifold if it satisfies $J{\mathcal{D}}_p^{\perp }=T_p^{\perp }M,\forall p\in M$.

Let M be a submanifold of $\tilde{M}$. For a unit vector $X\in T_p^1\left(M\right)$, the angle $\theta \left(X\right)$ between $JX$ and $T_{p}M$ is called the Wirtinger angle of X. A submanifold M of $\tilde{M}$ is called a slant or (θ slant) if $\theta \left(X\right)$ is independent of the choice of $X\in T_{p}M$ and of $p\in M$ (see [25]). In this case, $\theta$ is called the slant angle of M. Obviously, complex and totally real submanifolds are slant submanifolds with $\theta =0$ and $\theta =\frac{\pi }{2}$, respectively. A slant submanifold is called proper if $\theta \in (0,\frac{\pi }{2})$ (for results on slant submanifolds, we refer to [26,27,28,29,30,31,32,33] and the references therein, among many others).

For any tangent vector X of a submanifold M in $(\tilde{M},J,g)$, we put

$$\begin{array}{c}\hfill JX=PX+FX,\end{array}$$

(3)

where $PX$ and $FX$ denote the tangential and normal components of $JX$, respectively. A proper slant submanifold of a Kähler manifold is called a Kählerian slant [26,34] if P is parallel with respect to the Levi–Civita connection ∇ of M, i.e., $\nabla P=0$.

2.4. H-Umbilical Submanifolds

The notion of the H-umbilical submanifold was introduced in [35,36] as follows. A Lagrangian submanifold M of an almost Hermitian manifold is called H-umbilical if its second fundamental form satisfies

$$\begin{array}{cc}\hfill & h(E_1,E_1)=\lambda J{E}_1,h(E_2,E_2)=\cdots =h(E_n,E_n)=\mu J{E}_1,\hfill \\ & h(E_1,E_j)=\mu J{E}_j,h(E_j,E_k)=0,2\le j\ne k\le n,n=dimM\hfill \end{array}$$

(4)

for some functions $\lambda$ and $\mu$ with respect to an orthonormal local frame $\{E_1,\cdots ,E_n\}$. In this case, the ratio $\lambda :\mu$ is called the ratio of the H-umbilical submanifold M.

3. The First Chen Inequality for Submanifolds in Real Space Forms

The first inequality involving $\delta$-invariants is the following.

Theorem 2

([12,37]). Let M be a submanifold of a Riemannian manifold $R^m\left(c\right)$ of constant sectional curvature c. If $dimM=n\ge 3$, then

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c.\end{array}$$

(5)

The equality sign in (5) holds if, and only if, with respect to suitable orthonormal frame fields $\{e_1,\dots ,e_n,e_{n+1},\dots ,e_m\}$, the shape operator of M in $R^m\left(c\right)$ takes the following forms:

$$\begin{array}{cc}\hfill & A_{n+1}=\left(\begin{array}{ccccc}a& 0& 0& \cdots & 0\\ 0& \mu -a& 0& \cdots & 0\\ 0& 0& \mu & \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & \mu \end{array}\right),A_r=\left(\begin{array}{ccccc}{h}_{11}^r& h_{12}^r& 0& \cdots & 0\\ h_{12}^r& -h_{11}^r& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right),r=n+2,\dots ,m,\hfill \end{array}$$

(6)

where $A_s=A_{e_s},s=n+1,\dots ,m$.

Remark 1.

I. Mihai and R.-I. Mihai provide an alternate simpler proof of Theorem 2 in [38].

Assumption 1.

From now on, we always assume that submanifolds are of the dimension $n\ge 3$ unless mentioned otherwise.

4. $\mathit{\delta }\left(\mathbf{2}\right)$-Ideal Submanifolds of Real Space Forms

Definition 1.

A submanifold $M^n$ of a real space form is called $\delta \left(2\right)$-ideal if it satisfies the equality case of (5) identically.

Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is an immersion that produces the least possible amount of tension from the ambient space, see, e.g., [9,13,18,39].

It follows from Theorem 2 that there exists a canonical distribution $\mathcal{E}$ for n-dimensional $\delta \left(2\right)$-ideal submanifolds in $R^m\left(c\right)$ defined by

$$\begin{array}{c}\hfill \mathcal{E}\left(p\right)=\{X\in T_{p}N:(n-1)h(X,Y)=n〈X,Y〉H,\forall Y\in T_{p}N\}.\end{array}$$

(7)

Since 1993, many articles exist studying $\delta \left(2\right)$-ideal submanifolds in real space forms via condition (6). In this section, we present many results in this respect.

4.1. $\delta \left(2\right)$-Ideal Submanifolds in Euclidean Spaces

First, we gave the following result.

Theorem 3

([12]). If M is an n-dimensional minimal submanifold of ${\mathbb{E}}^m$, then M is $\delta \left(2\right)$-ideal if, and only if, locally, M is one of the following:

(1)

A totally geodesic submanifold of ${\mathbb{E}}^m$;

(2)

A spherical cylinder $\mathbb{R}\times S^{n-1}\left(c\right)$;

(3)

A direct product of a Euclidean k-space ${\mathbb{E}}^{\ell }$ and a Riemannian $(n-\ell )$-manifold $P^{n-2}$ satisfying $\delta \left(2\right)=0$ and $M={\mathbb{E}}^k\times P^{n-2}\subset {\mathbb{E}}^m$ is immersed as a minimal direct product submanifold.

Theorem 3 shows that minimal $\delta \left(2\right)$-ideal submanifolds of ${\mathbb{E}}^m$ are either affine subspaces or rotation hypersurfaces obtained by rotating a straight line, that is, cones and cylinders. In [40], M. Dajczer and L. A. Florit determined $\delta \left(2\right)$-ideal rotation submanifolds over surfaces in ${\mathbb{E}}^{n+1}$.

For $\delta \left(2\right)$-ideal minimal submanifolds of ${\mathbb{E}}^m$, we have the following:

Theorem 4

([12]). Let M be a minimal submanifold of ${\mathbb{E}}^m$ with $dimM=n$. If M is $\delta \left(2\right)$-ideal, then we have the following:

(1)

For a given point $p\in M$, we have either $dim\mathcal{E}\left(p\right)=n$ or $dim\mathcal{E}\left(p\right)=n-2$;

(2)

If $dim\mathcal{E}\equiv n$, then M is totally geodesic in ${\mathbb{E}}^m$;

(3)

If $dim\mathcal{E}\equiv n-2$, M is an $(n-2)$-ruled submanifold. In particular, if M is normal $(n-2)$-ruled, then M is an open portion of one the following two types of submanifolds:

(3.1)

The direct product $N\times {\mathbb{E}}^{n-2}$ of a linear $(n-2)$-subspace ${\mathbb{E}}^{n-2}\subset {\mathbb{E}}^m$ and a minimal surface N in ${\mathbb{E}}^{m-n+2}$;

(3.2)

The direct product $CM\times {\mathbb{E}}^{n-3}$ of a linear $(n-3)$-subspace ${\mathbb{E}}^{n-3}\subset {\mathbb{E}}^m$ and a three-dimensional minimal cone $CM$ in ${\mathbb{E}}^{m-n+3}$.

Conversely, every minimal submanifold of ${\mathbb{E}}^m$ given by $\left(3.1\right)$ or $\left(3.2\right)$ is $\delta \left(2\right)$-ideal.

A submanifold is called CMC if it has a constant mean curvature.

Theorem 5

([41]). A $CMC$ hypersurface M of ${\mathbb{E}}^{n+1}$ with $n>2$ is $\delta \left(2\right)$-ideal if, and only if, M is either minimal or an open portion of a spherical hypercylinder $\mathbb{R}\times S^{n-1}\left(r\right)\subset {\mathbb{E}}^{n+1}$.

M. Dajczer and L. A. Florit extended Theorem 5 to the following.

Theorem 6

([40]). An n-dimensional $CMC$ submanifold M of ${\mathbb{E}}^m$ with $n\ge 4$ is $\delta \left(2\right)$-ideal if, and only if, M is either minimal or an open portion of a spherical hypercylinder $\mathbb{R}\times S^{n-1}\left(r\right)\subset {\mathbb{E}}^{n+1}\subset {\mathbb{E}}^m$.

For conformally flat $\delta \left(2\right)$-ideal hypersurfaces, we have the following.

Theorem 7

([42]). If M is a $\delta \left(2\right)$-ideal hypersurface of ${\mathbb{E}}^{n+1}$, then M is a conformally flat manifold with a null scalar curvature if, and only if, M is totally geodesic.

For $\delta \left(2\right)$-ideal tubular hypersurfaces, we have the following.

Theorem 8

([43]). A tube $T^r\left(B^k\right)$ in ${\mathbb{E}}^{n+1}$ is $\delta \left(2\right)$-ideal if, and only if, $k=1$ and $B^k$ is an open part of a line. Thus, $T^r\left(B^k\right)$ is an open part of a spherical hypercylinder $\mathbb{R}\times S^{n-1}\left(r\right)$.

A submanifold M of a Euclidean space is said to be biharmonic (see, e.g., [21,44,45,46]) if its position vector field $\mathbf{x}$ satisfies ${\Delta }^2\mathbf{x}=0$, where $\Delta$ denotes the Laplacian of M.

A well-known biharmonic conjecture of the first author states that every biharmonic submanifold of a Euclidean space is minimal.

The next result is due to the first author and M. I. Munteanu.

Theorem 9

([47]). Every $\delta \left(2\right)$-ideal biharmonic hypersurface of ${\mathbb{E}}^{n+1}$ is minimal.

Remark 2.

Some further properties of $\delta \left(2\right)$-ideal hypersurfaces in Euclidean spaces have been obtained by many authors (see, e.g., [48,49], among others mentioned in book [18]).

Remark 3.

For the geometry of symmetries on $\delta \left(2\right)$-ideal hypersurfaces of a Euclidean space, see [48,49,50,51,52].

4.2. $\delta \left(2\right)$-Ideal Submanifolds in Real Space Forms

For $\delta \left(2\right)$-ideal submanifolds in real space forms, we also have the following.

Theorem 10

([42,53]). If M is a $\delta \left(2\right)$-ideal submanifold in a real space form $R^m\left(c\right)$, then M is Einstein if, and only if, M is totally geodesic.

Theorem 11

([41]). If M is a $CMC$ hypersurface in $S^{n+1}\left(1\right)$, then M is $\delta \left(2\right)$-ideal if, and only if, either one of the following applies:

(1)

M is totally geodesic;

(2)

There exist an open dense subset $W\subset M$ and a nontotally geodesic isometric minimal immersion $\psi :B^2\to S^{n+1}\left(1\right)$ of a surface $B^2$, such that W is an open subset of the unit normal bundle $N{B}^2$ given by $N_p{B}^2=\left\{\xi \in T_{\psi \left(p\right)}{S}^{n+1}\left(1\right):〈\xi ,{\psi }_{★}\left(T_p{B}^2\right)〉=0and〈\xi ,\xi 〉=1\right\}$.

Theorem 12

([41]). If M is a $CMC$ hypersurface in $H^{n+1}(-1)(n>2)$, then M is $\delta \left(2\right)$-ideal if, and only if, one of the following cases occurs.

(1)

M is totally geodesic;

(2)

M is a tube $T^r\left(B^2\right)$ with the radius $r={coth}^{-1}\left(\sqrt{2}\right)$ over a totally geodesic surface $B^2\subset H^{n+1}(-1)$;

(3)

There exist an open dense subset $W\subset M$ and a nontotally geodesic minimal immersion $\psi :B^2\to S_1^{n+1}$ of a surface $B^2$ into $S_1^{n+1}\left(1\right)$, such that W is an open subset of the unit normal bundle $N{B}^2$ of $B^2$ given by

$$N_p{B}^2=\left\{\xi \in T_{\psi \left(p\right)}{S}_1^{n+1}\left(1\right):\langle \xi ,{\psi }_{★}\left(T_p{B}^2\right)\rangle =0and〈\xi ,\xi 〉=-1\right\}.$$

Remark 4.

The complete classification of conformally flat $\delta \left(2\right)$-ideal hypersurfaces of real space forms was achieved by the first author and J. Yang in [41] (see also [54]).

The next two results classified $\delta \left(2\right)$-ideal tubular hypersurfaces in $S^{n+1}$ and $H^{n+1}$.

Theorem 13

([43]). Let $B^k$ be a k-dimensional submanifold of $S^{n+1}\left(1\right)$. Then, a tube $T^r\left(B^k\right)$ over $B^k$ is $\delta \left(2\right)$-ideal if, and only if, either (1) $r=\frac{\pi }{2}$ and $k=0$ or (2) $r=\frac{\pi }{2}$, $k=2$ and $B^2$ is a minimal surface in $S^{n+1}\left(1\right)$.

Theorem 14

([43]). If $T^r\left(B^k\right)$ is a tube over an embedded submanifold $B^k$ in $H^{n+1}(-1)$, then $T^r\left(B^k\right)$ is $\delta \left(2\right)$-ideal if, and only if, $r={coth}^{-1}\left(\sqrt{2}\right)$, $k=2$ and $B^k$ is totally geodesic in $H^{n+1}(-1)$.

For $\delta \left(2\right)$-ideal isoparametric hypersurfaces in real space forms, we have the following.

Theorem 15

([43]). If M is an isoparametric hypersurface of a complete simply connected real space form $R^{n+1}\left(c\right),c\in \{-1,0,1)$, then M is $\delta \left(2\right)$-ideal if, and only if, M is either a totally geodesic hypersurface or a tube $T^r\left(B^k\right)$ given by one of the following:

(1)

$c=0$, $k=1$ and $B^1$ is geodesic, i.e., M is locally $\mathbb{R}\times S^{n-1}\left(a\right)\subset {\mathbb{E}}^{n+1}$;

(2)

$c=1,n=3,k=2,r=\frac{\pi }{2}$ and $B^2$ is locally a Veronese surface in $S^4\left(1\right)$;

(3)

$c=-1,k=2,cothr=\sqrt{2}$ and $B^2$ is totally geodesic.

Remark 5.

For the classification of $\delta \left(2\right)$-ideal submanifolds with the type number $\le 2$ in real spaces, see [55].

5. Extensions of $\mathit{\delta }\left(\mathbf{2}\right)$ and First Inequality for Submanifolds in Riemannian Manifolds

For a natural number $n\ge 3$ and an integer $k\ge 0$, let $\mathcal{S}\left(n\right)$ be the set consisting of all k-tuples $(n_1,\dots ,n_k)$ of integers $2\le n_1,\dots ,n_k\le n-1$ with $n_1+\cdots +n_k\le n$.

The invariant $\delta \left(2\right)$ was extended in [13,56] to a series of Riemannian invariants as follows. For each given k-tuple $(n_1,\dots ,n_k)\in \mathcal{S}\left(n\right)$, the first author defined the $\delta$-invariant $\delta (n_1,\dots ,n_k)$ by

$$\begin{array}{cc}\hfill & \delta (n_1,\dots ,n_k)=\tau -inf\{\tau \left(L_1\right)+\cdots +\tau \left(L_k\right)\},\hfill \end{array}$$

(8)

where $L_1,\dots ,L_k$ run over all k mutually orthogonal subspaces of $T_{p}M$ with $dim{L}_j=n_j,j=1,\dots ,k$. In particular, we have the following:

• $\delta (\varnothing )=\tau$ ($k=0$, the trivial δ-invariant);
• $\delta \left(2\right)=\tau -infK,$ where K is the sectional curvature;
• $\delta (n-1)=maxRic$.

Note that $\delta \left(2\right)$ is the nontrivial first $\delta$-invariant in the series of $\delta$-invariants defined in (8).

Put

$$c(n_1,\dots ,n_k)=\frac{n^2(n+k-1-{\sum }_{j=1}^k{n}_j)}{2(n+k-{\sum }_{j=1}^k{n}_j)},b(n_1,\dots ,n_k)=\frac{n(n-1)}{2}-\frac{1}{2}\sum _{j=1}^k{n}_j(n_j-1).$$

To provide further solutions to Problem 1, we extended Theorem 2 to the following.

Theorem 16

([57]). Let $\varphi :M\to \tilde{M}$ be an isometric immersion between Riemannian manifolds. Then, we have

$$\begin{array}{cc}\hfill & \delta (n_1,\dots ,n_k)\le c(n_1,\dots ,n_k){\parallel H\parallel }^2+b(n_1,\dots ,n_k)max\tilde{K},\hfill \end{array}$$

(9)

where $max{\tilde{K}}_p$ denotes the maximum sectional curvature of $\tilde{M}$ restricted to two-plane sections of the tangent space $T_{p}M$.

In particular, we have the following (see, e.g., [13,18,39,56]).

Theorem 17.

Let $\varphi :M\to \tilde{M}\left(c\right)$ be an isometric immersion from a Riemannian n-manifold into a real space form of a constant curvature c. Then,

$$\begin{array}{cc}\hfill & \delta (n_1,\dots ,n_k)\le c(n_1,\dots ,n_k){\parallel H\parallel }^2+b(n_1,\dots ,n_k)c.\hfill \end{array}$$

(10)

Remark 6.

In 1999, B. D. Suceavă [58] proved the following result. Let M be an n-dimensional submanifold of a Riemannian manifold $\tilde{M}$. Then, the scalar curvature τ of M satisfies

$$\begin{array}{c}\hfill \tau \le \frac{n(n-1)}{2}{\parallel H\parallel }^2+\sum _{1\le i<j\le n}\tilde{K}(e_i,e_j),\end{array}$$

(11)

where $\tilde{K}(e_i,e_j)$ denotes the sectional curvature of $\tilde{M}$ restricted to the plane section spanned by $\{e_i,e_j\}$ for an orthonormal basis $\{e_1,\dots ,e_n\}$. Further, Suceavă constructed in [59], on an open subset of a multiwarped product of hyperbolic planes, a metric with a negative Ricci curvature that does not admit any minimal isometric immersion into a Euclidean space in any codimension via δ-invariants.

Remark 7.

Following (11), some authors provide different versions of the first Chen inequality replacing $max\tilde{K}$ in (9) for $\delta \left(2\right)$ using $\tilde{\tau }\left(M\right)-\tilde{K}\left(\pi \right)$, where $\tilde{\tau }\left(M\right)$ and $\tilde{K}\left(\pi \right)$ denote the scalar curvature and sectional curvature of a plane section $\pi \subset TM$ of the ambient space $\tilde{M}$ (see, e.g., [60]).

6. Submanifolds of Real Space Forms Equipped with a Nonsymmetric Connection

The notion of semisymmetric metric connections on a Riemannian manifold was introduced by H. A. Hayden in [61] as follows. A linear connection $\tilde{\nabla }$ on a Riemannian manifold $(M^m,g)$ is called a semisymmetric connection if the torsion tensor $\tilde{T}$ of $\tilde{\nabla }$ given by $\tilde{T}(\tilde{X},\tilde{Y})={\tilde{\nabla }}_{\tilde{X}}\tilde{Y}-{\tilde{\nabla }}_{\tilde{Y}}\tilde{X}-[\tilde{X},\tilde{Y}]$ for all vector fields $\tilde{X},\tilde{Y}\in \mathfrak{X}\left(M^m\right)$ satisfies

$$\tilde{\tilde{T}}(\tilde{X},\tilde{Y})=\omega \left(\tilde{Y}\right)\tilde{X}-\omega \left(\tilde{X}\right)\tilde{Y}$$

(12)

for some one-form $\omega$. Let $\tilde{P}$ be the vector field dual to the one-form $\omega$, i.e., $\omega \left(\tilde{X}\right)=g(\tilde{X},\tilde{P}).$ It was proved by K. Yano in [62] that a Riemannian manifold is conformally flat if, and only if, it admits a semisymmetric metric connection whose curvature tensor vanishes identically.

A semisymmetric connection $\tilde{\nabla }$ is said to be semisymmetric metric if $\tilde{\nabla }g=0$ holds identically. Following [62], a semisymmetric metric connection $\tilde{\nabla }$ on a Riemannian manifold $M^m$ is given by

$${\tilde{\nabla }}_{\tilde{X}}\tilde{Y}={\tilde{\nabla }}_{\tilde{X}}^{\circ }\tilde{Y}+\omega \left(\tilde{Y}\right)\tilde{X}-g(\tilde{X},\tilde{Y})\tilde{P},$$

where ${\tilde{\nabla }}^{\circ }$ is the Levi–Civita connection of $(\tilde{M},g)$ with respect to the Riemannian metric g.

Let $M^n$ be an n-dimensional submanifold of a real space form $R^m\left(c\right)$ of a constant curvature c and ${\tilde{R}}^{\circ }$ the curvature tensor of $R^m\left(c\right)$ with respect to the Levi–Civita connection. Then, the curvature tensor $\tilde{R}$, with respect to the semisymmetric metric connection $\tilde{\nabla }$ on $R^m\left(c\right)$, is given by [63]

$$\begin{array}{cc}\hfill \tilde{R}(X,Y,Z,W)=& {\tilde{R}}^{\circ }(X,Y,Z,W)-\alpha (Y,Z)g(X,W)+\alpha (X,Z)g(Y,W)\hfill \\ & -\alpha (X,W)g(Y,Z)+\alpha (Y,W)g(X,Z)\hfill \end{array}$$

(13)

for vector fields $X,Y,Z,W\in \mathfrak{X}\left(M^n\right)$, where $\alpha$ is a $(0,2)$-tensor field defined by

$$\begin{array}{c}\hfill \alpha (X,Y)=\left({\tilde{\nabla }}_X^{\circ }\omega \right)Y-\omega \left(X\right)\omega \left(Y\right)+\frac{1}{2}\omega \left(\tilde{P}\right)g(X,Y).\end{array}$$

Denote by $\widehat{H}$ and $H^{\circ }$ the mean curvature vector of $M^n$ with respect to the semisymmetric connection and with respect to the Levi–Civita connection on $R^m\left(c\right)$, respectively.

The following results were proved by A. Mihai and C. Özgür [64]).

Theorem 18.

Let $M^n$ be an n-dimensional submanifold of a real space form $R^m\left(c\right)$ of a constant sectional curvature c endowed with a semisymmetric metric connection $\tilde{\nabla }$. Then,

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel \widehat{H}\parallel }^2+(n-1)\frac{c}{2}-\lambda \right]+\mathrm{Trace}\left({\alpha |}_{{\pi }^{\perp }}\right),\hfill \end{array}$$

(14)

where π is a plane section of $TM$ and ${\pi }^{\perp }$ is the orthogonal complement of π.

Proposition 1.

The mean curvature $\widehat{H}$ coincides with the mean curvature $\mathring{H}$ of $M^n$ if, and only if, the vector field $\tilde{P}$ is tangent to $M^n$.

Theorem 19.

If the vector field $\tilde{P}$ is tangent to $M^n$, then the equality case of inequality (14) holds at point $p\in M^n$ if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_p{M}^n$ and an orthonormal basis $\{e_{n+1},\dots ,e_m\}$ of $T_p^{\perp }{M}^n$, such that the shape operator of $M^n$ in $R^m\left(c\right)$ at p satisfies (6).

Remark 8.

A. Mihai and C. Özgür also derived the first Chen inequality for the submanifold of a real space form $R^m\left(c\right)$ endowed with a semisymmetric nonmetric connection in [65]. See also [66] by L. Zhang and P. Zhang.

Remark 9.

N. Poyraz and H. I. Yoldaş derived in [67] the first Chen inequalities for submanifolds of real space forms endowed with a Ricci quarter-symmetric metric connection.

7. Submanifolds of Quasi-Real Space Forms

According to the first author and K. Yano [68], a Riemannian manifold $(\tilde{M},g)$ is said to be a quasi-real space form (or of quasi-constant curvature) if there is a unit vector field $\mathcal{P}$ and functions $\kappa$ and $\mu$ so that the curvature tensor $\tilde{R}$ of $\tilde{M}$ satisfies

$$\begin{array}{cc}\hfill \tilde{R}(X,Y,Z,W)& =\kappa \left[g\right(Y,Z\left)g\right(X,W)-g(X,Z\left)g\right(Y,W\left)\right]\hfill \\ & +\mu \left[g\right(X,W\left)\zeta \right(Y\left)\zeta \right(Z)-g(X,Z\left)\zeta \right(Y\left)\zeta \right(W)\hfill \\ & +g(Y,Z)\zeta \left(X\right)\zeta \left(W\right)-g(Y,W)\zeta \left(X\right)\zeta \left(Z\right)],\hfill \end{array}$$

where $\zeta$ is the one-form dual to a unit vector field $\mathcal{P}$. Let us denote such a manifold by ${\tilde{M}}_{\kappa ,\mu }^m\left(\zeta \right)$. For a submanifold M of ${\tilde{M}}_{\kappa ,\mu }^m\left(\zeta \right)$, let

$$\begin{array}{c}\hfill \mathcal{P}={\mathcal{P}}^T+{\mathcal{P}}^{\perp },\end{array}$$

(15)

where ${\mathcal{P}}^T$ and ${\mathcal{P}}^{\perp }$ are the tangential and normal components of $\mathcal{P}$. For a plane section $\pi$ in $T_{p}M,p\in M$, we put ${\mathcal{P}}_{\pi }=p{r}_{\pi }\mathcal{P}.$

C. Özgür investigated the first Chen inequality for submanifolds in a quasi-real space form ${\tilde{M}}_{\kappa ,\mu }^m\left(\zeta \right)$. He proved the following.

Theorem 20

([69]). If M is a submanifold of a quasi-real space form ${\tilde{M}}_{\kappa ,\mu }^m\left(\zeta \right)$, then

$$\begin{array}{cc}\hfill \delta \left(2\right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel H\parallel }^2+(n+1)\frac{\kappa }{2}\right]+\mu \left[(n-1)\parallel {\mathcal{P}}^T{\parallel }^2-{\parallel {\mathcal{P}}_{\pi }\parallel }^2\right],\hfill \end{array}$$

(16)

where $n=dimM$ and π is a plane section of $T_{p}M$ at point $p\in M$.

Corollary 1

([69]). Under the same assumptions as in Theorem 18, we have the following:

(a) 

If $\mathcal{P}$ is tangent to M, then

$$\begin{array}{cc}\hfill \delta \left(2\right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel H\parallel }^2+(n+1)\frac{\kappa }{2}\right]+\mu \left[(n-1)-\parallel {\mathcal{P}}_{\pi }{\parallel }^2\right].\hfill \end{array}$$

(b) 

$\mathcal{P}$ is normal to M, then

$$\begin{array}{cc}\hfill \delta \left(2\right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel H\parallel }^2+(n+1)\frac{\kappa }{2}\right].\hfill \end{array}$$

In [69], C. Özgür also determined the necessary and sufficient conditions for inequality (11) to be an equality.

Remark 10.

In 2016, P. Zhang, L. Zhang and W. Song [70] extended Theorem 2 from submanifolds of a real space form to submanifolds of a quasi-real space form. Consequently, they extended Theorem 20 from $\delta \left(2\right)$ to $\delta (n_1,\dots ,n_k)$ as well.

In 2009, U. C. De and A. K. Gazi [71] defined a Riemannian manifold of a nearly quasi-constant curvature as a Riemannian manifold whose curvature tensor satisfies

$$\begin{array}{cc}\hfill \tilde{R}(X,Y,Z,W)& =\kappa \left[g\right(Y,Z\left)g\right(X,W)-g(X,Z\left)g\right(Y,W\left)\right]\hfill \\ & -\mu \left[g\right(X,W\left)B\right(Y,Z)-g(X,Z\left)B\right(Y,W)\hfill \\ & +g(Y,Z)B(X,W)-g(Y,W)B(X,Z)],\hfill \end{array}$$

where $\kappa ,\mu$ are functions and B is a nonzero symmetric tensor of type $(0,2)$.

C. Özgür and A. De investigated the first Chen inequality for submanifolds in a nearly quasi-real space form. They obtained the following.

Theorem 21

([72]). If M is a submanifold with $dim=n$ in a nearly quasi-real space form, then

$$\begin{array}{cc}\hfill \delta \left(2\right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel H\parallel }^2+(n-1)\frac{\kappa }{2}\right]+\mu \left[(n-2)\lambda +\mathrm{Trace}\left(B{|}_{{\pi }^{\perp }}\right)\right],\hfill \end{array}$$

where π is a plane section of $T_{p}M(p\in M)$ and $\lambda =\mathrm{Trace}\left(B\right)$.

Remark 11.

Some other versions of the first Chen-type inequality for submanifolds of nearly quasi-constant curvature manifolds were also obtained in [73,74].

8. Submanifolds of Quasi-Real Space Forms with a Nonsymmetric Connection

In this section, we follow the notations from Section 6 and Section 7. For submanifolds of a quasi-real space form endowed with a nonsymmetric connection, P. Zhang, L. Zhang and W. Song proved the following.

Theorem 22

([75]). Let $M^n$ be a submanifold of a quasi-real space form ${\tilde{M}}_{\kappa ,\mu }^m\left(\zeta \right)$ endowed with a semisymmetric metric connection. Then, we have

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel \widehat{H}\parallel }^2+(n+1)\frac{\kappa }{2}\right]-(n-2)\lambda \hfill \\ & +\mu \left[(n-1)\parallel {\mathcal{P}}^T{\parallel }^2-{\parallel {\mathcal{P}}_{\pi }\parallel }^2\right]-\mathrm{Trace}\left({\alpha |}_{{\pi }^{\perp }}\right),\hfill \end{array}$$

(17)

where π is a plane section of $TM$ and ${\pi }^{\perp }$ is the orthogonal complement of π.

Theorem 22 implies the following.

Corollary 2.

Under the same assumptions as in Theorem 22, we have the following:

(a) 

If $\mathcal{P}$ is tangent to M, then

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & (n-2)\left[\frac{n^2}{2(n-1)}{\parallel \widehat{H}\parallel }^2+(n+1)\frac{\kappa }{2}\right]+\mu \left[(n-1)-\parallel {\mathcal{P}}_{\pi }{\parallel }^2\right]\hfill \\ & -(n-2)\lambda -\mathrm{Trace}\left({\alpha |}_{{\pi }^{\perp }}\right).\hfill \end{array}$$

(b) 

$\mathcal{P}$ is normal to M, then

$$\tau -K\left(\pi \right)\le (n-2)\left[\frac{n^2}{2(n-1)}{\parallel \widehat{H}\parallel }^2+(n+1)\frac{\kappa }{2}\right]-(n-2)\lambda -\mathrm{Trace}\left({\alpha |}_{{\pi }^{\perp }}\right).$$

Remark 12.

In [75], P. Zhang, L. Zhang and W. Song also extended Theorem 2 from submanifolds of real space forms to submanifolds of quasi-real space forms endowed with a nonsymmetric connection. Consequently, they also extended Theorem 22 from $\delta \left(2\right)$ to $\delta (n_1,\dots ,n_k)$.

By a nearly quasi-real space form we mean a Riemannian manifold whose curvature tensor satisfies (see [71])

$$\begin{array}{c}\tilde{R}(X,Y,Z,W)=\kappa \{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\}\\ -\mu \left\{g\right(X,W\left)B\right(Y,Z)-g(X,Z\left)B\right(Y,W)+g(Y,Z\left)B\right(X,W)-g(Y,W\left)B\right(X,Z\left)\right\},\end{array}$$

where $\kappa ,\mu$ are functions and B is a nonzero symmetric tensor of type $(0,2)$. Let us denote such a nearly quasi-real space form by ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$. Assume that ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$ can be endowed with a nonsymmetric connection $\overline{\nabla }$. Let us consider a $(0,2)$-tensor field s on ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$ defined by

$$\begin{array}{c}\hfill s(\overline{X},\overline{Y})=\left({\overline{\nabla }}_{\overline{X}}^{\circ }\omega \right)\overline{Y}-\omega \left(\overline{X}\right)\omega \left(\overline{Y}\right),\end{array}$$

(18)

where $\omega$ is defined by (12) and ${\overline{\nabla }}^{\circ }$ is the Levi–Civita connection of ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$.

For a submanifold M of a nearly quasi-real space form ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$ equipped with a nonsymmetric connection $\overline{\nabla }$, put

$$\begin{array}{cc}\hfill & \chi =\mathrm{Trace}B,\lambda =\mathrm{Trace}s,\mathrm{\Omega }\left(X\right)=s(X,X)+g({\mathcal{P}}^{\perp },h(X,X))\hfill \end{array}$$

(19)

for a unit vector X tangent to M.

The following result was proved by P. Zhang, X. Pan and L. Zhang.

Theorem 23

([76]). Let M be a submanifold of a Riemannian manifold of nearly quasi-constant curvature ${\widehat{N}}_{\kappa ,\mu }\left(B\right)$. Then, for any $X\in T^{1}M$, we have

$$\begin{array}{c}\tau -K(\pi )\le \frac{(n+1)(n-2)}{2}\kappa +\mu \left[{(n-2)\chi +\mathrm{Trace}B|}_{{\pi }^{\perp }}\right]+\frac{n^2(n-2)}{2(n-1)}{\parallel \widehat{H}\parallel }^2\\ -\frac{n-1}{2}\lambda -\frac{n(n-1)}{2}\omega \left(H\right)+\mathrm{\Omega }\left(X\right),\end{array}$$

(20)

where π is a plane section and λ is the trace of B. The equality case of (20) holds at point $p\in M$ if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_p{M}^n$ and an orthonormal basis $\{e_{n+1},\dots ,e_m\}$ of $T_p^{\perp }M$, such that the shape operator of M at p satisfies (6).

9. General Submanifolds in Complex Space Forms and Applications

Let ${\tilde{M}}^m\left(c\right)$ denote a complex m-dimensional Kähler manifold of a constant holomorphic sectional curvature c. Then, the Riemann curvature tensor of ${\tilde{M}}^m\left(c\right)$ satisfies

$$R(X,Y)Z=\frac{c}{4}[g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY+2g(X,JY)JZ].$$

By a complex space form, we mean a Kähler manifold of a constant holomorphic sectional curvature. A complete simply connected complex space form ${\tilde{M}}^m\left(c\right)$ is biholomorphic isometric to a complex projective space $C{P}^m\left(c\right)$, a complex Euclidean space ${\mathbb{C}}^m$ or a complex hyperbolic space $C{H}^m\left(c\right)$ according to $c>0$, $c=0$ or $c<0$, respectively.

By applying the same proof of Theorem 2, we have the following.

Theorem 24

([77]). If $:M^n$ is a submanifold in a complex space form ${\tilde{M}}^m\left(4c\right)$, then, for any point $p\in M^n$ and any plane section $\pi \subset T_p{M}^n$, we have

$$\begin{array}{c}\hfill \tau -K\left(\pi \right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c+\frac{3}{2}{\parallel P\parallel }^{2}c-3\mathrm{\Psi }\left(\pi \right)c,\end{array}$$

(21)

where P is the canonical endomorphism defined by (3) and $\mathrm{\Psi }\left(\pi \right)={〈P{e}_1,e_2〉}^2$ with $\{e_1,e_2\}$ being an orthonormal basis of π.

The equality in (21) holds at $p\in M^n$ if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{p}M$ and an orthonormal basis $\{e_{n+1},\dots ,e_{2m}\}$ of $T_p^{\perp }M$, such that $\left(a\right)$$\pi =\mathit{Span}\{e_1,e_2\}$ and $\left(b\right)$ the shape operators $A_r=A_{e_r}$, $r=n+1,\dots ,2m,$ take the following forms:

$$\begin{array}{cc}\hfill & A_{n+1}=\left(\begin{array}{ccccc}a& 0& 0& \dots & 0\\ 0& b& 0& \dots & 0\\ 0& 0& \mu & \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & \mu \end{array}\right),A_r=\left(\begin{array}{ccccc}{h}_{11}^r& h_{12}^r& 0& \dots & 0\\ h_{12}^r& -h_{11}^r& 0& \dots & 0\\ 0& 0& 0& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 0\end{array}\right)\hfill \end{array}$$

(22)

for $\mu =a+b$ and $r=n+2,\dots ,2m$.

The next two results follow easily from Theorem 24.

Theorem 25

([77]). Let $M^n$ be a submanifold of a complex projective m space $C{P}^m\left(4c\right)$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n^2+2n-2)c,c>0.\end{array}$$

(23)

The equality in (23) holds identically if, and only if, n is even and $M^n$ is immersed as a holomorphic, totally geodesic submanifold of $C{P}^m\left(4c\right)$.

Theorem 26

([77]). Let $M^n$ be a submanifold of a complex hyperbolic m space $C{H}^m\left(4c\right)$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c,c<0.\end{array}$$

(24)

The equality in (24) holds at point $p\in M^n$ if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_p{M}^n$ and an orthonormal basis $\{e_{n+1},\dots ,e_{2m}\}$ of $T_p^{\perp }{M}^n$, such that (a) the subspace spanned by $e_3,\dots ,e_n$ is totally real, (b) $K(e_1\wedge e_2)=infK$ at p and (c) the shape operators of $M^n$ take the form of (22).

Remark 13.

Inequality (21) was extended in [78,79] to submanifolds of complex space forms endowed with a semisymmetric metric connection.

10. Totally Real and Lagrangian Submanifolds of Complex Space Forms

Since the proof of Theorem 2 is based only on Gauss’ equation, the same proof of Theorem 2 can be used to obtain the following.

Theorem 27.

If $M^n$ is a totally real (or Lagrangian) submanifold of a complex space form ${\tilde{M}}^m\left(4c\right)$, then

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c.\end{array}$$

(25)

The equality in (25) holds at point $p\in M^n$ if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_p{M}^n$ and an orthonormal basis $\{e_{n+1},\dots ,e_{2m}\}$ of $T_p^{\perp }{M}^n$, such that the shape operators of $M^n$ take the form of (21).

Remark 14.

The equality case of (25) is known as the first Chen equality for totally real submanifolds in complex space forms.

Remark 15.

In [80], D. E. Blair, F. Dillen, L. Verstraelen and L.Vrancken constructed many totally real and minimal $\delta \left(2\right)$-ideal immersions of $S^3$ into $C{P}^{2n+1}\left(4\right)$ via Calabi curves.

In 1994, the first author, F. Dillen, L. Verstraelen and L. Vrancken proved the following.

Theorem 28

([81]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(c\right)$. If $M^n$ satisfies the equality in (25), then the mean curvature vector H satisfies $〈H,JX〉=0$ for any tangent vector $X\in T{M}^n$.

Theorem 28 implies the following.

Theorem 29

([81]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(c\right)$. If it satisfies the equality in (25), then it is minimal.

The first author, F. Dillen, L. Verstraelen and L. Vrancken also proved, in 1996, the following.

Theorem 30

([82]). Let $M^n$ be a Lagrangian submanifold of a complex space form $\tilde{M}\left(c\right),c\in \{-1,0,1\}$. If $M^n$ has a constant scalar curvature, then $M^n$ is $\delta \left(2\right)$-ideal if, and only if, either (1) $M^n$ is totally geodesic in $\tilde{M}\left(c\right)$ or (2) $n=3$, $c=1$ and the immersion is locally congruent to a given equivariant immersion of $S^3\to \mathbb{C}{P}^3$ of a topological three-sphere $S^3$ with a homogeneous metric.

$\delta \left(2\right)$-ideal Lagrangian submanifolds of $C{H}^n(-4)$ with integrable ${\mathcal{E}}^{\perp }$ were classified by the first author and L. Vrancken.

Theorem 31

([83]). Assume that $f:M\to C{H}^n(-4)$ is a $\delta \left(2\right)$-ideal Lagrangian immersion without geodesic points. If ${\mathcal{E}}^{\perp }$ is integrable, then, up to rigid motions, point p of an open dense subset of M has a neighborhood $U_p$, such that the immersion F is given by one of the following:

(a) 

$f(t,x,y)=\pi (cosht(\varphi \left(x\right),0,\dots ,0)+sinht(0,0,0,\psi \left(y\right)\left)\right),$ where

$$\psi :(y_1,\dots ,y_{n-3})\mapsto \psi \left(y\right)$$

describes the standard totally real $(n-3)$-sphere $S^{n-3}$ in ${\mathbb{E}}^{n-2}\subset {\mathbb{C}}^{n-2}$ and $\varphi :(x,y)\mapsto \varphi \left(x\right)$ is a minimal horizontal immersion in $H_1^5(-1)$;

(b) 

$f=\pi (cosht(\psi \left(y\right),0,0,0\left)\right)-sinht(0,\dots ,0,\varphi (x\left)\right))$, where

$$\psi :(y_1,\dots ,y_{n-3})\mapsto \psi \left(y\right)$$

describes the standard totally real hyperbolic space $H_1^{n-3}(-1)$ lying in ${\mathbb{E}}_1^{n-2}\subset {\mathbb{C}}_1^{n-2}$ and $\varphi :(x_1,x_2)\mapsto \psi \left(x\right)$ is a minimal horizontal immersion in $S^5\left(1\right)$;

(c) 

$f(t,x,y)=\pi ((cosht,-sinht,0,\dots ,0)+\frac{e^{-t}}{2}z(x,y)(1,-1,0,\dots ,0)+\frac{1}{2}{e}^{-t}(0,0,w_1\left(x\right),$

$w_2\left(x\right),y_1,\dots ,y_{n-3}\left)\right),$ where

$$w:D\subset {\mathbb{R}}^2\to {\mathbb{C}}^2:(x_1,x_2)\mapsto (w_1(x_1,x_2),w_2(x_1,x_2))$$

is a minimal Lagrangian immersion and z is a complex-valued function determined by

$$2(z+\overline{z})=w_1{\overline{w}}_1+w_2{\overline{w}}_2+\sum _{i=1}^{n-3}{v}_i^2,$$

and by the condition that its imaginary part depends only on x and satisfies the following system:

$$\begin{array}{cc}& 2{(z-\overline{z})}_{x_1}=w_1{\left({\overline{w}}_1\right)}_{x_1}+w_2{\left({\overline{w}}_2\right)}_{x_1}-{\overline{w}}_1{\left(w_1\right)}_{x_1}-{\overline{w}}_2{\left(w_2\right)}_{x_1},\hfill \end{array}$$

$$\begin{array}{c}\hfill 2{(z-\overline{z})}_{x_2}=w_1{\left({\overline{w}}_1\right)}_{x_2}+w_2{\left({\overline{w}}_2\right)}_{x_2}-{\overline{w}}_1{\left(w_1\right)}_{x_2}-{\overline{w}}_2{\left(w_2\right)}_{x_2},\end{array}$$

and $\pi :H_1^{2m+1}(-1)\to C{H}^m(-4)$ is a generalized Hopf fibration.

Remark 16.

$\delta \left(2\right)$-ideal Lagrangian submanifolds in $C{H}^3(-4)$ with nonintegrable ${\mathcal{E}}^{\perp }$ were also determined in [83].

11. ${\mathit{\delta }}_{\mathit{k}}^{\mathit{r}}\left(\mathbf{2}\right)$-Invariant for Kähler Submanifolds in Complex Space Forms

Let M be a Kähler manifold of complex dimension $n\ge 2$. A plane section $\pi \subset T_{p}M,p\in M,$ is called totally real if $J\pi$ is orthogonal to $\pi$. Denote by $K\left({\pi }^r\right)$ the sectional curvature of a totally real plane section ${\pi }^r$. Let $(inf{K}^r)\left(p\right)={inf}_{{\pi }^r\subset T_{p}M}K\left({\pi }^r\right),$ where $K\left({\pi }^r\right)$ runs over all totally real plane sections at $p\in M$. For each $k\in \mathbb{R}$, the first author defined in [84] a totally real δ-invariant${\delta }_k^r\left(2\right)$ on M by

$$\begin{array}{c}\hfill {\delta }_k^r\left(p\right)=\tau \left(p\right)-kinf{K}^r\left(p\right),p\in M,\end{array}$$

(26)

where $inf{K}^r\left(p\right)={inf}_{{\pi }^r}\left\{K\left({\pi }^r\right)\right\}$ and ${\pi }^r$ runs over all totally real plane sections in $T_{p}M$.

It is well-known that the shape operator of a Kähler submanifold M in a Kähler manifold ${\tilde{M}}^{n+p}$ satisfies

$$A_{\alpha }=\left(\begin{array}{cc}{A}_{\alpha }^{\prime }& A_{\alpha }^{″}\\ A_{\alpha }^{″}& -A_{\alpha }^{\prime }\end{array}\right),A_{{\alpha }^{*}}=\left(\begin{array}{cc}-A_{\alpha }^{″}& A_{\alpha }^{\prime }\\ A_{\alpha }^{\prime }& A_{\alpha }^{″}\end{array}\right).$$

(27)

Condition (27) implies that every Kähler submanifold in a Kähler manifold ${\tilde{M}}^{n+p}$ is minimal, i.e., $\mathrm{Trace}{A}_{\alpha }=\mathrm{Trace}{A}_{\alpha *}=0,\alpha =1,\dots ,p$.

The first author proved in [84] the following.

Theorem 32.

If M is a Kähler submanifold of complex dimension $n\ge 2$ in a complex space form ${\tilde{M}}^m\left(4c\right)$, then the following applies:

(a) 

For each real number $k\in (-\infty ,4]$, ${\delta }_k^r$ satisfies

$$\begin{array}{c}\hfill {\delta }_k^r\le (2n^2+2n-k)c;\end{array}$$

(28)

(b) 

Inequality (28) fails for every $k>4$;

(c) 

${\delta }_k^r=(2n^2+2n-k)c$ holds identically for some $k\in (-\infty ,4)$ if, and only if, M is totally geodesic;

(d) 

${\delta }_4^r=(2n^2+2n-4)c$ holds at $p\in M$ if, and only if, there is an orthonormal basis

$$e_1,\dots ,e_n,e_{1^{*}}=J{e}_1,\dots ,e_{n^{*}}=J{e}_n,{\xi }_1,\dots ,{\xi }_{m-n},{\xi }_{1^{*}}=J{\xi }_{,}\dots ,{\xi }_{{(m-n)}^{*}}=J{\xi }_{m-n}$$

of $T_p{\tilde{M}}^m\left(4c\right)$, such that

$$\begin{array}{c}{A}_{\alpha }=\left(\begin{array}{cc}{A}_{\alpha }^{\prime }& A_{\alpha }^{″}\\ A_{\alpha }^{″}& -A_{\alpha }^{\prime }\end{array}\right),A_{{\alpha }^{*}}=\left(\begin{array}{cc}-A_{\alpha }^{″}& A_{\alpha }^{\prime }\\ A_{\alpha }^{\prime }& A_{\alpha }^{″}\end{array}\right),\hfill \\ A_{\alpha }^{\prime }=\left(\begin{array}{cc}\begin{array}{cc}{a}_{\alpha }& b_{\alpha }\\ b_{\alpha }& -a_{\alpha }\end{array}& 0\\ 0& 0\end{array}\right),A_{\alpha }^{”}=\left(\begin{array}{cc}\begin{array}{cc}{a}_{\alpha }^{*}& b_{\alpha }^{*}\\ b_{\alpha }^{*}& -a_{\alpha }^{*}\end{array}& 0\\ 0& 0\end{array}\right)\hfill \end{array}$$

(29)

for some $n\times n$ matrices $A_{\alpha }^{\prime },A_{\alpha }^{″}$, with respect to this basis.

Definition 2

([84]). A Kähler submanifold N of a Kähler manifold $M^m$ is called strongly minimal if $A_{\alpha }^{\prime }$ and $A_{\alpha }^{″}$ in (27) satisfy $\mathrm{Trace}{A}_{\alpha }^{\prime }=\mathrm{Trace}{A}_{\alpha }^{″}=0,\mathrm{for}\alpha =1,\dots ,m-n.$

The next two results are also due to [84].

Theorem 33.

A complete Kähler submanifold M of complex dimension $n\ge 2$ in $C{P}^m\left(4\right)$ satisfies ${\delta }_4^r=2(n^2+n-2)$ identically if, and only if, either M is a totally geodesic Kähler submanifold or $n=2$ and M is a strongly minimal Kähler surface in $C{P}^m\left(4\right)$.

Theorem 34.

A complete Kähler submanifold M with ${dim}_{\mathbb{C}}M=n\ge 2$ of ${\mathrm{C}}^m$ satisfies ${\delta }_4^r=0$ identically if, and only if, either M is a complex n plane of ${\mathrm{C}}^m$ or M is a complex cylinder over a strongly minimal Kähler surface.

Clearly, every totally geodesic Kähler submanifold of a complex space form is strongly minimal. The following provides some nontrivial examples of strongly minimal Kähler surfaces. Next, we provide some nontotally geodesic examples of strongly minimal Kähler surfaces.

Example 1.

The complex quadric $Q_2=\{(z_0,z_1,z_2,z_3)\in C{P}^3\left(4\right):z_0^2+z_1^2+z_2^2+z_3^2=0\}$ is a strongly minimal surface in $C{P}^3\left(4\right)$, where $\{z_0,z_1,z_2,z_3\}$ is a homogeneous coordinate system of $C{P}^3\left(4\right)$.

Example 2.

The Kähler surface $\{(z_1,z_2,z_3)\in {\mathbb{C}}^3:z_1^2+z_2^2+z_3^2=1\}$ is a strongly minimal surface in ${\mathbb{C}}^3$.

The next example is due to B. D. Suceavă.

Example 3

([85,86]). The Kähler surface $\{(z_1,z_2,z_3)\in {\mathbb{C}}^3:z_1+z_2+z_3^2=k\},k\in \mathbb{C},$ is a strongly minimal surface in ${\mathbb{C}}^3$.

Remark 17.

For further results on strongly minimal complex submanifolds of Kähler manifolds, see [85,86].

Remark 18.

In [87], M. Gülbahar and E. Kılıç studied strongly minimal complex light-like hypersurfaces of an indefinite Kähler manifold.

12. Improved Inequalities for Lagrangian Submanifolds in Complex Space Forms

In 2007, using an optimization technique, T. Oprea improved inequality (25) to the following.

Theorem 35

([88]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(c\right)$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(2n-3)}{2(2n+3)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c.\end{array}$$

(30)

Remark 19.

Inequality (30) was known as Oprea’s improved first Chen inequality.

In 2009, J. Bolton, C. Rodriguez Montealegre and L.Vrancken proved the following result in [89].

Theorem 36.

Let $M^n$ be a Lagrangian submanifold of $\mathbb{C}{P}^n\left(4\right)$ attaining equality (30) at every point. If $n\ge 4$, then $M^n$ is minimal. Hence, $M^n$ is one of the submanifolds of $\mathbb{C}{P}^n\left(4\right)$ discussed in Theorem 30.

In [90], J. Bolton and L. Vrancken studied nonminimal three-dimensional Lagrangian submanifolds of $C{P}^3\left(4\right)$ satisfying equality (30) at every point. Their main result shows that each such submanifold can be constructed from a certain minimal Lagrangian surface in $C{P}^2\left(4\right)$.

A complete classification of nonminimal Lagrangian submanifolds of complex space forms ${\tilde{M}}^3\left(4c\right)$ attaining equality (30) at every point was achieved, in 2012, by the first author, F. Dillen and L. Vrancken in [91].

L. Vrancken studied $\delta \left(2\right)$-ideal submanifolds from a global point of view. He proved the following.

Theorem 37

([92]). Let M be a $\delta \left(2\right)$-ideal complete Lagrangian immersion of the complex projective space $C{P}^n\left(4\right)$. Then, either M is totally geodesic or $n=3$.

The next two optimal inequalities for Lagrangian submanifolds were proved by the first author, F. Dillen, J. Van der Veken and L. Vrancken in 2013.

Theorem 38

([93]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(4c\right)$. If $(n_1,\dots ,n_k)\in \mathcal{S}\left(n\right)$ satisfies $n_1+\dots +n_k<n$, then, at any point of $M^n$, we have

$$\begin{array}{cc}\delta (n_1,\dots ,n_k)\le & \frac{n^2\left(n-{\sum }_{i=1}^k{n}_i+3k-1-6{\sum }_{i=1}^k\frac{1}{2+n_i}\right)}{2\left(n-{\sum }_{i=1}^k{n}_i+3k+2-6{\sum }_{i=1}^k\frac{1}{2+n_i}\right)}{\parallel H\parallel }^2\hfill \\ & +\frac{1}{2}\left(n(n-1)-\sum _{i=1}^k{n}_i(n_i-1)\right)c.\hfill \end{array}$$

(31)

Remark 20.

Inequality (30) is the special case of (31) with $k=1$ and $n_1=2$.

Theorem 39

([93]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(4c\right)$. If $(n_1,\dots ,n_k)\in \mathcal{S}\left(n\right)$ satisfies $n_1+\dots +n_k=n$, then, at any point of $M^n$, the following holds:

$$\begin{array}{cc}& \delta (n_1,\dots ,n_k)\le \frac{n^2\left(k-1-2{\sum }_{i=2}^k\frac{1}{n_i+2}\right)}{2\left(k-2{\sum }_{i=2}^k\frac{1}{n_i+2}\right)}{\parallel H\parallel }^2+\frac{1}{2}\left(n(n-1)-\sum _{i=1}^k{n}_i(n_i-1)\right)c.\hfill \end{array}$$

(32)

Remark 21.

The equality cases of (31) and (32) have been characterized in [93].

The next result of T. Oprea is a consequence of Theorem 38.

Corollary 3

([94]). Let $M^n$ be a Lagrangian submanifold of a complex space form ${\tilde{M}}^n\left(4c\right)$. Then,

$$\delta (n-1)\le \frac{n-1}{4}\left({n\parallel H\parallel }^2+4c\right).$$

(33)

Remark 22.

The classification of nonminimal Lagrangian submanifolds of ${\tilde{M}}^n\left(4c\right)$ satisfying the equality case of inequality (33) was also achieved in [91] by the first author, F. Dillen and L. Vrancken (see also [95] by S. Deng).

13. Slant Submanifolds of Complex Space Forms

As an application of Theorem 24, A. Oiagă and I. Mihai [96] gave the following (see also [97,98]).

Theorem 40.

Let M be an n-dimensional θ-slant submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$. Then, we have

$$\delta \left(2\right)\le \frac{n-2}{2}\left\{\frac{n^2}{n-1}{\parallel H\parallel }^2+\frac{1}{4}(n+1+3{cos}^2\theta )c\right\}.$$

(34)

The equality in (34) holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal frame.

For n-dimensional Kählerian slant submanifolds in a complex space form ${\tilde{M}}^n\left(4c\right)$, A. Mihai proved the following improved first Chen inequality.

Theorem 41

( [99]). Let M be an n-dimensional Kählerian slant submanifold of a complex space form ${\tilde{M}}^n\left(4c\right)$ and $\pi \subset T_{p}M$ a two-plane section at $p\in M$. Then, we have

$$\begin{array}{c}\hfill \tau \left(p\right)-K\left(\pi \right)\le \frac{n^2(2n-3)}{2(2n+3)}{∥H∥}^2+[(n+1)(n-2)+3n{cos}^2\theta -6\mathrm{\Phi }\left(\pi \right)]\frac{c}{2}.\end{array}$$

(35)

Moreover, the equality case of inequality (35) holds at a point $p\in M$ if, and only if, there exists an orthonormal basis $\{e_1,e_2,\dots ,e_n\}$ at p, such that, with respect to this basis, the second fundamental form takes the following form:

$$\begin{array}{cc}& h(e_1,e_1)=a{e}_1^{*}+3b{e}_3^{*}h(e_1,e_3)=3b{e}_1^{*}h(e_3,e_j)=4b{e}_j^{*}\hfill \\ & h(e_2,e_2)=-a{e}_1^{*}+3b{e}_3^{*}h(e_2,e_3)=3b{e}_2^{*}h(e_j,e_k)=4b{e}_3^{*}{\delta }_{jk}\hfill \\ & h(e_1,e_2)=-a{e}_2^{*}h(e_3,e_3)=12b{e}_3^{*}h(e_1,e_j)=h(e_2,e_j)=0,\end{array}$$

for some numbers $a,b$ and $j,k=4,\dots ,n$, where $e_i^{*}=(csc\theta )F{e}_i,i=1,\dots ,n$.

14. CR-Submanifolds of Complex Space Forms

For $CR$-submanifolds of complex space forms, we have the following sharp inequalities involving $\delta \left(2\right)$.

Theorem 42

([39,77]). Let M be an n-dimensional $CR$-submanifold in a complex space form ${\tilde{M}}^m\left(4c\right)$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \left\{\begin{array}{cc}{{\displaystyle \frac{n^2(n-2)}{2(n-1)}}\parallel H\parallel }^2+\left\{{\displaystyle \frac{1}{2}}(n+1)(n-2)+3\mathfrak{h}\right\}c,\hfill & \mathit{if}c>0;\hfill \\ {{\displaystyle \frac{n^2(n-2)}{2(n-1)}}\parallel H\parallel }^2,\hfill & \mathit{if}c=0;\hfill \\ {{\displaystyle \frac{n^2(n-2)}{2(n-1)}}\parallel H\parallel }^2+{\displaystyle \frac{1}{2}}(n+1)(n-2)c,\hfill & \mathit{if}c<0,\hfill \end{array}\right.\end{array}$$

where $\mathfrak{h}$ denotes the complex dimension of the holomorphic distribution $\mathcal{D}$.

For a $CR$-submanifold M of a Kähler manifold, the first author introduced the CR-δ-invariant ${\delta }_{\mathcal{D}}\left(2\right)$ by

$$\begin{array}{c}\hfill {\delta }_{\mathcal{D}}\left(2\right)=\tau -\tau \left(\mathcal{D}\right),\end{array}$$

where τ and $\tau \left(\mathcal{D}\right)$ denote the scalar curvature of M and the scalar curvature of the complex distribution $\mathcal{D}\subset TM$, respectively (see [100,101]). For a CR-submanifold M, the two partial mean curvature vectors $H_{\mathcal{D}}$ and $H_{{\mathcal{D}}^{\perp }}$ of M are defined by

$$\begin{array}{cc}\hfill & H_{\mathcal{D}}=\frac{1}{2\mathfrak{h}}\sum _{i=1}^{2\mathfrak{h}}\sigma (e_i,e_i),H_{{\mathcal{D}}^{\perp }}=\frac{1}{\mathfrak{p}}\sum _{r=2\mathfrak{h}+1}^{2\mathfrak{h}+\mathfrak{p}}\sigma (e_r,e_r),\hfill \end{array}$$

where $\mathfrak{h}={\mathrm{rank}}_{\mathbb{C}}\mathcal{D}$ and $\mathfrak{p}=\mathrm{rank}{\mathcal{D}}^{\perp }$.

The next two results were proved by F. R. Al-Solamy, the first author and S. Deshmukh.

Theorem 43

([102]). Let M be an antiholomorphic submanifold of a complex space form ${\tilde{M}}^{\mathfrak{h}+\mathfrak{p}}\left(4c\right)$ with $\mathfrak{h}={\mathrm{rank}}_{\mathbb{C}}\mathcal{D}\ge 1$ and $\mathfrak{p}=\mathrm{rank}{\mathcal{D}}^{\perp }\ge 2$. Then, we have

$$\begin{array}{c}\hfill {\delta }_{\mathcal{D}}\left(2\right)\le \frac{{(2\mathfrak{h}+\mathfrak{p})}^2}{2}{\parallel H\parallel }^2+{\displaystyle \frac{\mathfrak{p}}{2}}(4\mathfrak{h}+\mathfrak{p}-1)c-\frac{3{\mathfrak{p}}^2}{2(\mathfrak{p}+2)}{\parallel H_{{\mathcal{D}}^{\perp }}\parallel }^2.\end{array}$$

(36)

The equality sign of (36) holds identically if, and only if, the following three conditions are satisfied:

(a) 

M is $\mathcal{D}$-minimal, i.e., $H_{\mathcal{D}}=0$;

(b) 

M is mixed totally geodesic;

(c) 

There exist an orthonormal frame $\{e_1,\dots ,e_{2\mathfrak{h}}\}$ of $\mathcal{D}$ and an orthonormal frame $\{e_{2\mathfrak{h}+1},\dots ,e_n\}$ of ${\mathcal{D}}^{\perp }$, such that the second fundamental h of M satisfies

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{h}_{rr}^r=3h_{ss}^r,\mathrm{for}2\mathfrak{h}+1\le r\ne s\le 2\mathfrak{h}+\mathfrak{p},\hfill \\ h_{st}^r=0,\mathrm{for}\mathrm{distinct}r,s,t\in \{2\mathfrak{h}+1,\dots ,2\mathfrak{h}+\mathfrak{p}\}.\hfill \end{array}\right.\hfill \end{array}$$

Example 4

(Whitney sphere). Let $w:S^{\mathfrak{p}}\left(1\right)\to {\mathbb{C}}^{\mathfrak{p}},\mathfrak{p}\ge 2,$ be the map given by

$$w(y_0,y_1,\dots ,y_{\mathfrak{p}})=\frac{1+\mathrm{i}{y}_0}{1+y_0^2}(y_1,\dots ,y_{\mathfrak{p}}),y_0^2+y_1^2+\dots +y_{\mathfrak{p}}^2=1.$$

Then, w gives rise to a (nonisometric) Lagrangian immersion with one self-intersection point. This immersion is called the Whitney $\mathfrak{p}$-sphere (see, e.g., [18]).

Theorem 44

([102]). Let M be an antiholomorphic submanifold in a complex space form ${\tilde{M}}^{1+\mathfrak{p}}\left(4c\right)$ with $\mathfrak{h}={\mathrm{rank}}_{\mathbf{C}}\mathcal{D}=1$ and $p=\mathrm{rank}{\mathcal{D}}^{\perp }\ge 2$. Then, we have

$$\begin{array}{c}\hfill {\delta }_{\mathcal{D}}\left(2\right)\le {\displaystyle \frac{(\mathfrak{p}-1){(\mathfrak{p}+2)}^2}{2(\mathfrak{p}+2)}}{H}^2+{\displaystyle \frac{\mathfrak{p}}{2}}(\mathfrak{p}+3)c.\end{array}$$

(37)

The equality case of (37) holds identically if, and only if, $c=0$ and either one of the following occurs:

(i) 

M is a totally geodesic antiholomorphic submanifold of ${\mathbb{C}}^{h+p}$;

(ii) 

Up to dilations and rigid motions, M is given by an open portion of the following product immersion:

$$\begin{array}{c}\hfill \varphi :\mathbb{C}\times S^p\left(1\right)\to {\mathbf{C}}^{1+\mathfrak{p}};(z,x)\mapsto (z,w\left(x\right)),z\in \mathbb{C},x\in S^{\mathfrak{p}}\left(1\right),\end{array}$$

where $w:S^{\mathfrak{p}}\left(1\right)\to {\mathbb{C}}^{\mathfrak{p}}$ is the Whitney $\mathfrak{p}$-sphere.

In 1999, the first author and L. Vrancken classified $\delta \left(2\right)$-ideal proper CR-submanifolds in $C{H}^m(-4)$ as follows.

Theorem 45

([103]). Let U be a domain of $\mathbb{C}$ and $\varphi :U\to {\mathbb{C}}^{m-1}$ be a nonconstant holomorphic curve in ${\mathbb{C}}^{m-1}$. Define $z:{\mathbb{R}}^2\times U\to {\mathbb{C}}_1^{m-1}$ by

$$z(t,w)=\left(-1-\frac{1}{2}\varphi \left(w\right)\overline{\varphi }\left(w\right)+iu.-\frac{1}{2}\varphi \left(w\right)\overline{\varphi }\left(w\right)+iu,\varphi \left(w\right)\right)e^{it}.$$

Then, $〈z,z〉=-1$ and the image $z({\mathbb{R}}^2\times U)$ in $H_1^{2m+1}(-1)$ is invariant under the group action of $H_1^1=\{\lambda \in \mathbb{C}:\lambda \overline{\lambda }=1\}$. Moreover, away from points where ${\varphi }^{\prime }\left(w\right)=0$, the quotient space $\pi (z{\left(\mathbb{R}\right)}^2\times U)$ is a $\delta \left(2\right)$-ideal proper CR-submanifold of $C{H}^m(-4)$, where $\pi :H_1^{2m+1}\to C{H}^m(-4)$ denotes the Hopf fibration.

Conversely, up to rigid motions $C{H}^m(-4)$, every $\delta \left(2\right)$-ideal proper CR-submanifold of $C{H}^m(-4)$ is obtained in such way.

Remark 23.

For further results on submanifolds of complex space forms involving δ-invariants, see [104,105].

15. Submanifolds of Generalized Complex Space Forms

15.1. Generalized Complex Space Forms

An RK-manifold is an almost Hermitian manifold $(\tilde{M},J,g)$ whose curvature tensor $\tilde{R}$ satisfies $\tilde{R}(JX,JY,JZ,JW)=\tilde{R}(X,Y,Z,W).$ For an RK-manifold, put $\lambda (X,Y)=\tilde{R}(X,Y,JX,JY)-\tilde{R}(X,Y,X,Y).$ An almost Hermitian manifold $(\tilde{M},J,g)$ is called a pointwise constant type if at any $p\in \tilde{M}$ and any $X\in T_p\tilde{M}$, we have $\lambda (X,Y)=\lambda (X,Z),$ where $Y,Z\in T_p^1\left(\tilde{M}\right)$ are perpendicular to X and $JX$. Further, $\tilde{M}$ is said to be a constant type if $\lambda (X,Y)$ is a constant function for any $X,Y\in T^1\left(M\right)$ satisfying $g(X,Y)=g(JX,Y)=0$.

An $RK$-manifold of a constant holomorphic sectional curvature and constant type is called a generalized complex space form.

It is easy to see that a complex space form is a generalized complex space form, but the converse is not true; the simplest such example is the nearly Kähler unit six-sphere $S^6\left(1\right)$.

Let $\tilde{M}(c,\alpha )$ denote a generalized complex space form of constant holomorphic sectional curvature c and constant type α. The curvature tensor $\tilde{R}$ of $\tilde{M}(c,\alpha )$ satisfies

$$\begin{array}{cc}\hfill \tilde{R}(X,Y)Z=& \frac{c+3\alpha }{4}[g(Y,Z)X-g(X,Z)Y]\hfill \\ & +\frac{c-\alpha }{4}[g(X,JZ)JY-g(Y,JZ)JX+2g(X,JY)JZ].\end{array}$$

Obviously, a generalized complex space form $\tilde{M}(c,\alpha )$ is a real space form if $c=\alpha$; it is a complex space form of constant holomorphic sectional curvature c if $\alpha =0$.

15.2. The First Chen Inequality for Submanifolds of Generalized Complex Space Forms

For submanifolds of a generalized complex space form, J.-S. Kim, Y.-M. Song and M. M. Tripathi proved the following.

Theorem 46

([106]). If M is an n-dimensional submanifold of a generalized complex space form $\tilde{M}(c,\alpha )$, then, for each point $p\in M$ and each plane section $\pi \subset T_{p}M$, we have

$$\tau -K\left(\pi \right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{(n+1)(n-2)}{8}(c+3\alpha )+\frac{3(c-\alpha )}{8}{\parallel \mathcal{P}\parallel }^2-\frac{3(c-\alpha )}{4}\mathrm{\Psi }\left(\pi \right).$$

The equality holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to a suitable orthonormal frame.

Theorem 47

([106]). If M is an n-dimensional θ-slant submanifold of a generalized complex space form $\tilde{M}(c,\alpha )$, then we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n-2}{2}\left(\frac{n^2}{(n-1)}{\parallel H\parallel }^2+\frac{(n+1)(c+3\alpha )}{4}+\frac{3(c-\alpha )}{4}{cos}^2\theta \right).\end{array}$$

(38)

The equality in (38) holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal basis.

Remark 24.

Theorem 46 extended Theorem 24 from submanifolds of complex space forms to submanifolds of generalized complex space forms. Moreover, Theorems 24 and 46 were extended further to submanifolds of generalized complex space forms equipped with a semisymmetric metric connection by V. Ghişoiu in [107] (see also [108]).

Remark 25.

In [109], S. Sular derived the first Chen inequality for submanifolds of generalized space forms endowed with a semisymmetric metric connection.

15.3. Slant Submanifolds of Generalized Complex Space Forms

The first generalization of slant immersions was given by N. Papaghiuc [110], who defined the notion of a semi-slant submanifold as follows: A submanifold M of an almost Hermitian manifold is called semi-slant if the tangent bundle of M admits a direct orthogonal decomposition: $TM={\mathcal{D}}_1\oplus {\mathcal{D}}_2,$ where ${\mathcal{D}}_1$ a complex distribution and ${\mathcal{D}}_2$ is a slant one, i.e., the angle $\theta \left(X_p\right)$, $0<\theta \left(X_p\right)\le \pi /2$ between $J\left(X_p\right)$ and ${\mathcal{D}}_2$ is a constant.

On the other hand, A. Carriazo defined in [111] the notions of pseudo-slant submanifolds (also known as hemi-slant or anti-slant submanifolds) and bi-slant submanifolds. Both of them satisfy a decomposition $TM={\mathcal{D}}_1\oplus {\mathcal{D}}_2,$ of the tangent bundle with the following differences: for a pseudo-slant submanifold, ${\mathcal{D}}_1$ is a totally real distribution and distribution ${\mathcal{D}}_2$ is a slant distribution ${\mathcal{D}}_2$ with angle ${\theta }_2\ne \pi /2$, while a submanifold is said to be bi-slant if both distributions are slant, with angles ${\theta }_1,{\theta }_2\in [0,\pi /2]$. A bi-slant submanifold is called proper bi-slant if one has ${\theta }_1,{\theta }_2\in (0,\pi /2)$. For a bi-slant submanifold, let $d_i=\frac{1}{2}{dim}_{\mathbb{R}}{\mathcal{D}}_i$ and ${\theta }_i$ the slant angle of ${\mathcal{D}}_i$, $i=1,2$.

The next result was given in [106] and in [112].

Theorem 48

([106,112]). Let M be an n-dimensional θ-slant submanifold of a generalized complex space form $\tilde{M}(c,\alpha )$. Then, we have

$$\delta \left(2\right)\le \frac{n-2}{2}\left\{\frac{n^2}{n-1}{\parallel H\parallel }^2+\frac{n+1}{4}(c+3\alpha )+\frac{3(c-\alpha )}{4}{cos}^2\theta \right\}.$$

(39)

The equality in (39) holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to a suitable orthonormal frame.

16. Submanifolds of Locally Conformal Kähler Manifolds

A Hermitian manifold $(\tilde{M},J,g)$ is called a locally conformal Kähler manifold [113] if each point $p\in \tilde{M}$ has an open neighborhood U with function $\sigma :U\to \mathbb{R}$, such that $\tilde{g}=e^{-\sigma }{g|}_U$ is a Kähler metric. If we choose $U=\tilde{M}$, then $(\tilde{M},J,g)$ is called a globally conformal Kähler manifold. The fundamental two-form ω is defined as $\omega (X,Y)=g(JX,Y)$ for $X,Y\in \mathfrak{X}\left(\tilde{M}\right)$.

The next proposition provides a necessary and sufficient condition for a Hermitian manifold to be locally conformal Kähler (see [113]).

Proposition 2.

A Hermitian manifold $(\tilde{M},J,g)$ is a locally conformal Kähler manifold if, and only if, there is a global one-form α, such that

$$\left({\tilde{\nabla }}_Z\omega \right)(Y,X)=\alpha \left(X\right)g(Z,Y)-\alpha \left(Y\right)g(Z,X)-\alpha \left(JX\right)\omega (Z,Y)+\alpha \left(JY\right)\omega (Z,X)$$

for $X,Y,Z\in \mathfrak{X}\left(\tilde{M}\right)$.

The one-form α is called the Lee form. For a locally conformal Kähler manifold $(\tilde{M},J,g)$, one defines a symmetric $(0,2)$-tensor $\tilde{Q}$ by

$$\begin{array}{c}\hfill \tilde{Q}(Y,X)=-\left({\tilde{\nabla }}_Y\alpha \right)\left(X\right)-\alpha \left(Y\right)\alpha \left(X\right)+\frac{1}{2}{\parallel \alpha \parallel }^{2}g(Y,X),\end{array}$$

(40)

where $\parallel \alpha \parallel$ denotes the norm of the Lee form with respect to g. If the holomorphic sectional curvature of $(\tilde{M},J,g)$ is a real constant c, then $\tilde{M}$ is called a locally conformal Kähler space form, denoted by $\tilde{M}\left(c\right)$.

In [114], K. Matsumoto and I. Mihai derived a version of the first Chen inequality for totally real submanifolds in a locally conformal Kähler space form. In [115], S. Hong, K. Matsumoto and M. M. Tripathi extended Theorem 24 to the following inequality for totally real submanifolds in a locally conformal Kähler space form.

Theorem 49.

Let M be an n-dimensional submanifold of a locally conformal Kähler space form ${\tilde{M}}^m\left(c\right)$ of constant holomorphic sectional curvature c. Then, for each point $p\in M$ and each plane section $\pi =\mathrm{Span}\{e_1,e_2\}$, we have

$$\begin{array}{cc}\hfill & \tau -K\left(\pi \right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{8}(n+1)(n-2)c+\frac{3}{8}c{\parallel P\parallel }^2-\frac{3}{4}c{〈P{e}_1,e_2〉}^2\hfill \\ & +\frac{3}{4}\left[(n-1)\mathrm{Trace}\left(\tilde{Q}{|}_M\right)-\mathrm{Trace}\left(\tilde{Q}{|}_{\pi }\right)+\sum _{i,j=1}^n〈P{e}_i,e_j〉\tilde{Q}(e_i,J{e}_j)\right]-\frac{3}{2}〈P{e}_1,e_2〉\tilde{Q}(e_2,J{e}_2).\hfill \end{array}$$

(41)

The equality holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal frame $\{e_1,\dots ,e_n\}$ tangent to M and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m}\}$.

Remark 26.

For totally real submanifolds of ${\tilde{M}}^m\left(c\right)$, (41) reduces to the following:

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{8}(n+1)(n-2)c\hfill \\ & +\frac{3}{4}\left[(n-1)\mathrm{Trace}\left(\tilde{Q}{|}_M\right)-\mathrm{Trace}\left(\tilde{Q}{|}_{\pi }\right))\right].\end{array}$$

For CR-submanifolds of a locally conformal Kähler space form ${\tilde{M}}^m\left(c\right)$, D. W. Yoon obtained the following.

Theorem 50

([116]). Let M be an n-dimensional CR-submanifold of a locally conformal Kähler space form ${\tilde{M}}^m\left(c\right)$. Then, for each point $p\in M$ and each plane section $\pi \subset T_{p}M$, we have

$$\tau -K\left(\pi \right)\le \frac{n-2}{2}\left\{\frac{n^2}{n-1}{\parallel H\parallel }^2+\frac{1}{4}(n+1)(c+6\sigma )\right\}+\frac{3}{4}(c-2\sigma )d,$$

(42)

where $d={\mathrm{rank}}_{\mathbb{C}}\mathcal{D}$ and $\sigma =\frac{1}{n}{\sum }_{i=1}^n\tilde{Q}(e_i,e_i)$. Moreover, the equality in (42) holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal frame $\{e_1,\dots ,e_n\}$ tangent to M with $\pi =\mathrm{Span}\{e_1,e_2\}$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m}\}$.

17. $\mathit{\delta }\left(\mathbf{2}\right)$-Ideal Lagrangian Submanifolds of Nearly Kähler ${\mathit{S}}^{\mathbf{6}}\left(\mathbf{1}\right)$

A Kähler manifold is an almost Hermitian manifold $(\tilde{M},J,\tilde{g})$, such that the tensor field $G=\tilde{\nabla }J=0$, where $\tilde{\nabla }$ is the Levi–Civita connection. By definition, a nearly Kähler manifold is an almost Hermitian manifold, such that G satisfies

$$G(X,X)=\left({\tilde{\nabla }}_{X}J\right)X=0.$$

(43)

Since the Kähler condition $\tilde{\nabla }J=0$ is relaxed by condition (43), a non-Kähler nearly Kähler manifold is called strict. The first example of such a manifold, the unit sphere $S^6\left(1\right)$, was introduced by T. Fukami and S. Ishihara in [117]. Later on, these manifolds were thoroughly studied by A. Gray in the series of his papers. The case of $six$-dimensional strict nearly Kähler manifolds is of particular importance since they are building blocks of arbitrary nearly Kähler manifolds.

N. Ejiri [118] proved that every Lagrangian submanifold of the nearly Kähler $S^6\left(1\right)$ is minimal. Thus, by combining this result with the first Chen inequality (5), we obtain the following inequality:

$$\begin{array}{c}\hfill \delta \left(2\right)\le 2\end{array}$$

(44)

for each Lagrangian submanifold of the nearly Kähler $S^6\left(1\right)$. Now, by applying Theorem 2, we have the following.

Theorem 51

([18], Theorem 19.4). Let M be a Lagrangian submanifold of $S^6\left(1\right)$. Then, the equality in (44) holds if, and only if, there exists a tangent basis $\{e_1,e_2,e_3\}$, such that

$$\begin{array}{c}h(e_1,e_1)=-h(e_2,e_2)=\lambda J{e}_1,h(e_1,e_2)=-\lambda J{e}_2,\hfill \\ h(e_1,e_3)=h(e_2,e_3)=h(e_3,e_3)=0,\hfill \end{array}$$

where λ satisfies $2{\lambda }^2=3-\tau$ and τ is the scalar curvature.

The following result is due to R. Sharma.

Theorem 52

([119]). If a Lagrangian submanifold M of the nearly Kähler $S^6\left(1\right)$ satisfies the first Chen equality, i.e., $\delta \left(2\right)=2$, and is conformally flat, then it is totally geodesic.

For Lagrangian submanifolds of $S^6\left(1\right)$ with a constant scalar curvature, a complete classification of $\delta \left(2\right)$-ideal Lagrangian submanifolds has been obtained by the first author, F. Dillen, L. Verstraelen and L. Vrancken in [120].

The next result of the same four authors classified $\delta \left(2\right)$-ideal Lagrangian submanifolds in $S^6\left(1\right)$, such that the distribution $\mathcal{E}$ (defined by (7)) satisfies the following two conditions: (a) the dimension of $\mathcal{E}$ is constant and (b) ${\mathcal{E}}^{\perp }$ is an integrable distribution.

Theorem 53

([121]). Let $f:N^2\to S^6\left(1\right)$ be a nontotally geodesic, minimal Lagrangian immersion whose ellipse of curvature is a circle. Then, $N^2$ is linearly full in a totally geodesic $S^5\left(1\right)$. Let ξ be a unit vector perpendicular to this $S^5\left(1\right)$. Then,

$$x:\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\times N^2\to S^6\left(1\right),(t,p)\mapsto (sint)\xi +(cost)f\left(p\right)$$

is a $\delta \left(2\right)$-ideal Lagrangian immersion.

Conversely, every nontotally geodesic $\delta \left(2\right)$-ideal Lagrangian immersion of M into $S^6\left(1\right)$ satisfying conditions (a) and (b) can be locally obtained in this way.

The complete classification of $\delta \left(2\right)$-ideal Lagrangian submanifolds in the nearly Kähler $S^6\left(1\right)$ was achieved by F. Dillen and L. Vrancken. In fact, they obtained the following.

Theorem 54

([122]). The following results hold.

(1) 

If $\psi :N_1\to C{P}^2\left(4\right)$ is an almost complex curve in $C{P}^2\left(4\right)$, $P{N}_1$ is the circle bundle over $N_1$ induced by $\pi :S^5\left(1\right)\to C{P}^2\left(4\right)$ and $\varphi :P{N}_1\to S^5\left(1\right)$ is the isometric immersion, such that $\pi \circ \varphi =\psi \circ \pi$, then there is a totally geodesic embedding $i:S^5\left(1\right)\to S^6\left(1\right)$, such that the composition $i\circ \varphi :P{N}_1\to S^6\left(1\right)$ is a $\delta \left(2\right)$-ideal Lagrangian immersion.

(2) 

Suppose that $\overline{\psi }:N_2\to S^6\left(1\right)$ is an almost complex curve without totally geodesic points. If we denote by $T^1{N}_2$ the unit tangent bundle of $N_2$ and by h the second fundamental form of $\overline{\psi }$, and define a map

$$\begin{array}{c}\hfill \overline{\varphi }:T^1{N}_2\to S^6\left(1\right):v\mapsto {\overline{\psi }}_{★}\left(v\right)\times \frac{h(v,v)}{\parallel h(v,v)\parallel },\end{array}$$

then, $\overline{\varphi }$ is a (possibly branched)$\delta \left(2\right)$-ideal Lagrangian immersion. Moreover, the immersion is linearly full in $S^6\left(1\right)$.

(3) 

Let $\overline{\psi }:N_2\to S^6\left(1\right)$ be a (branched) almost complex immersion and let $S{N}_2$ be the tube of radius $\frac{\pi }{2}$ over $N_2$ in the direction ${(L_0\oplus L_1)}^{\perp }$, where $L_0$ is ${\overline{\psi }}_{*}\left(T{N}_2\right)$ and $L_1$ is the first normal space, except at the (isolated) branch points. Then, $S{N}_2$ is a $\delta \left(2\right)$-ideal (possibly branched) Lagrangian submanifold of $S^6\left(1\right)$.

(4) 

If $f:M\to S^6\left(1\right)$ is a Lagrangian immersion that is not linearly full in $S^6\left(1\right)$, then M satisfies $\delta \left(2\right)=2$ and there is a totally geodesic $S^5$ and an almost complex immersion $\psi :N_1\to C{P}^2\left(4\right)$, such that f is congruent to ψ, which is obtained from ψ as in (1).

(5) 

Suppose that $f:M\to S^6\left(1\right)$ is a linearly full $\delta \left(2\right)$-ideal Lagrangian immersion. Let p be a nontotally geodesic point in M. Then, there is a (possibly branched) almost complex curve $\overline{\psi }:N_2\to S^6\left(1\right)$, such that f is locally around p congruent to $\overline{\psi }$, which is obtained from $\overline{\psi }$ as in (3).

18. $\mathit{\delta }\left(\mathbf{2}\right)$-Ideal CR-Submanifolds of Nearly Kähler ${\mathit{S}}^{\mathbf{6}}\left(\mathbf{1}\right)$

A proper CR-submanifold of nearly Kähler $S^6\left(1\right)$ must be of dimensions three, four and five due to dimension restrictions. All hypersurfaces of the sphere are trivially CR-submanifolds; thus, the focus of the investigation is on CR-submanifolds of dimensions three and four. The classification of $\delta \left(2\right)$-ideal minimal CR-submanifolds was discussed by M. Djorić and L. Vrancken in [123].

The following result is due to M. Antić.

Theorem 55

([124]). Let M be a proper three-dimensional CR-submanifold of $S^6\left(1\right)$ ruled by $S^2\left(1\right)$. If $|cos\varphi |$ is the length of the projection of the unit normal to the leaf of ruling at a point on the almost complex distribution, then ϕ is constant. Moreover, M is locally congruent to the immersion:

(a) 

For $cos\varphi \ne 0$,

$$\begin{array}{cc}\hfill p(u,v,w)& =sin(v+h)\gamma \times {\gamma }^{\prime }\hfill \\ \hfill & +cos(v+h)(cosu\sigma +sinu(cos\varphi {\gamma }^{\prime }-sin\varphi \gamma \times {\gamma }^{\prime })\times \sigma ),\hfill \end{array}$$

where γ is a sphere curve, with the arc-length parameter w, such that $\langle \gamma \times {\gamma }^{\prime },{\gamma }^{″}\rangle =0$, σ is a sphere curve parameterized by w orthogonal to γ, ${\gamma }^{\prime }$, $\gamma \times {\gamma }^{\prime }$, such that

$$\begin{array}{cc}\hfill & \langle {\sigma }^{\prime },\gamma \times {\gamma }^{\prime }\rangle =\langle \sigma \times {\sigma }^{\prime },\gamma \rangle =0,\hfill \\ & \langle \sigma \times {\sigma }^{\prime },sin\varphi {\gamma }^{\prime }+cos\varphi \gamma \times {\gamma }^{\prime }\rangle =\frac{cos\varphi }{2},\hfill \\ \hfill & \langle \sigma ,cos\varphi (\gamma \times {\gamma }^{\prime })\times {\gamma }^{″}+sin\varphi {\gamma }^{\prime }\times {\gamma }^{″}\rangle =0,\hfill \end{array}$$

and h is a function of w, such that $cos(v+h)>0$.

(b) 

For $cos\varphi =0$,

$$\begin{array}{cc}\hfill & f(u,v,w)=cosucosv\gamma +sinucosv{A}_2+sinv\gamma \times A_2,\hfill \end{array}$$

where γ is a nonconstant sphere curve parameterized by w and $A_2$ is a vector field along γ orthogonal to $\gamma ,{\gamma }^{\prime }$ and $\gamma \times {\gamma }^{\prime }$.

Now, assume that M is a three-dimensional minimal proper CR-submanifold contained in a totally geodesic $S^5\left(1\right)\subset S^6\left(1\right)$. As a totally geodesic hypersphere is obtained by taking the intersection of $S^6\left(1\right)$ with a hyperplane through the origin, it follows that there exists a constant unit length vector field V, namely the unit normal to that plane, which is normal to the submanifold M and tangent to the sphere $S^6\left(1\right)$.

The next result is due to M. Antić and L. Vrancken.

Theorem 56

([125]). Let M be a minimal proper three-dimensional CR-submanifold of $S^6\left(1\right)$, which is not linearly full in $S^6\left(1\right)$. Then, M is locally congruent to the immersion

$$\begin{array}{cc}\hfill F(s,u,v)=& cosucosv\{cos\left({\mu }_{1}s\right)e_0+sin\left({\mu }_{1}s\right)e_4\}\hfill \\ & +sinucosv\{cos\left({\mu }_{2}s\right)e_1+sin\left({\mu }_{2}s\right)e_5\}\hfill \\ \hfill & +sinv\{cos\left({\mu }_{3}s\right)e_2+sin\left({\mu }_{3}s\right)e_6\},\hfill \\ & {\mu }_1+{\mu }_2+{\mu }_3=0,{\mu }_1^2+{\mu }_2^2+{\mu }_3^2\ne 0,\hfill \end{array}$$

where $e_0,\dots ,e_6$ is a standard $G_2$ basis of the space ${\mathbb{R}}^7$.

The following result was obtained by M. Antić, M. Djorić and L. Vrancken.

Theorem 57

([126]). Let M be a four-dimensional minimal $\delta \left(2\right)$-ideal CR-submanifold in $S^6\left(1\right)$. Then, M is locally congruent with the immersion

$$\begin{array}{cc}\hfill & f(s,u,v,w)=(coswcosscosucosv,sinwsinscosucosv,\hfill \\ \hfill & sin2wsinvcosu+cos2wsinu,0,sinwcosscosucosv,\hfill \\ \hfill & coswsinscosucosv,cos2wsinvcosu-sin2wsinu).\hfill \end{array}$$

Now, let us consider a four-dimensional CR-submanifold of $S^6\left(1\right)$ having its totally real distribution totally geodesic, with a two-dimensional nullity distribution that is locally congruent to the immersion

$$\begin{array}{cc}\hfill & F_1(y_1,y_2,y_3,y_4,s)=y_1\gamma \left(s\right)+y_2{A}_3\left(s\right)+y_3{A}_3\times {\gamma }^{\prime }\left(s\right)-y_4(\gamma \times {\gamma }^{\prime })\left(s\right),\hfill \end{array}$$

(45)

where $y_1^2+y_2^2+y_3^2+y_4^2=1$, γ is a unit length sphere curve and $A_3$ is a unit length vector field along γ orthogonal to $\gamma ,{\gamma }^{\prime }$ and $\gamma \times {\gamma }^{\prime }$, such that

$$\begin{array}{c}\hfill A_3^{\prime }\times A_3\perp \gamma ,\gamma \times {\gamma }^{\prime },A_3-\langle A_3^{\prime },{\gamma }^{\prime }\rangle \gamma \perp {\gamma }^{″}\times {\gamma }^{\prime }.\end{array}$$

(46)

The next theorem of M. Antić and L. Vrancken is a generalization of Theorem 57.

Theorem 58

([127]). Let M be a four-dimensional CR-submanifold of the sphere $S^6\left(1\right)$ admitting a two-dimensional nullity distribution. Then, it is locally congruent to immersion (45), with conditions (46), or to the immersion

$$\begin{array}{cc}\hfill & F(s,u,v,w)=sin(v+h)L+cos(v+h)(cos(u+\chi )(B_{1}cos{f}_1\hfill \\ \hfill & +\frac{sin{f}_1}{\sqrt{4+m^2}}(-2B_1+m(L\times B_1))\times L^{\prime })+sin(u+\chi )L^{\prime }),\hfill \end{array}$$

where h and χ are functions of s, such that $cos(u+\chi ),cos(v+h)>0$, m is a constant, $f_1$ is a function of w and s, such that ${\partial }_w{f}_1>0$, L is a sphere curve parameterized by its arc-length s, such that $\langle L^{″},L\times L^{\prime }\rangle =0$, $B_1$ is a unit vector field along L, orthogonal to $L,L^{\prime }$, $L\times L^{\prime }$, $L^{″}\times L$, such that $\langle B_1^{\prime }\times B_1,L\rangle =0$ and $\langle L^{″}\times L^{\prime },m{B}_1+2L\times B_1\rangle =0$.

Remark 27.

For further results on $\delta \left(2\right)$-ideal CR-submanifolds of the nearly Kähler $S^6\left(1\right)$, see [128,129].

19. $\mathit{\delta }\left(\mathbf{2}\right)$-Ideal Associative Submanifolds of ${\mathit{S}}^{\mathbf{7}}\left(\mathbf{1}\right)$

There is a standard cross product × on ${\mathbb{R}}^7$ related to the octonians. The 14-dimensional exceptional Lie group $G_2$ may be regarded as the automorphisms of ${\mathbb{R}}^7$ preserving ×, and a $G_2$-structure on a seven-dimensional manifold $M^7$ may be regarded as such a cross product on each tangent space. This induces both a Riemannian metric and an orientation on $M^7$, and this structure is called torsion-free if the cross product is parallel with respect to the Levi–Civita connection of $M^7$.

A three-dimensional submanifold of $M^7$ is called associative if its tangent spaces are closed under cross product ×. Associative submanifolds are calibrated [130]. Examples of associative submanifolds of $S^7\left(1\right)$ include Lagrangian submanifolds of nearly Kähler $S^6\left(1\right)$, Hopf lifts of holomorphic curves in $C{P}^3\left(4\right)$ and minimal Legendre submanifolds of the Sasakian $S^7\left(1\right)$.

The following results are due to J. D. Lotay.

Theorem 59

([131]). If S is an oriented real analytic surface in $S^7\left(1\right)$, then locally there exists a unique associative submanifold in $S^7\left(1\right)$, which contains S.

Theorem 60

([131]). $\delta \left(2\right)$-ideal associative submanifolds in $S^7\left(1\right)$ are in one-to-one correspondence with "linear" pseudo-holomorphic curves in the space $\mathcal{C}$ of oriented geodesic circles in $S^7\left(1\right)$.

Theorem 61

([131]). If M is an associative submanifold in $S^7\left(1\right)$, which is $\delta \left(2\right)$-ideal, then either one of the following applies:

(a) 

M is the Hopf lift of a holomorphic curve in $C{P}^3\left(4\right)$;

(b) 

M is constructed from an isotropic minimal surface $\mathrm{\Sigma }\subset S^6\left(1\right)$ and a holomorphic curve Γ in a $C{P}^1$-bundle over Σ, such that the pseudo-holomorphic lift of Σ in $\mathcal{C}$ is "linear".

Theorem 62

([131]). For a minimal two-sphere $S^2$ of a nonconstant curvature in $S^6\left(1\right)$, there exist a Riemannian $three$-manifold $(N,g_N)$, which is an $S^1$-bundle over $S^2$, and an $S^1$-family of noncongruent isometric associative embeddings of $(N,g_N)$ in $S^7\left(1\right)$, which are $\delta \left(2\right)$-ideal.

20. Inequality for Submanifolds in a Quaternionic Space Form

An almost quaternionic Hermitian manifold $(\overline{M},\mathrm{\Sigma },\overline{g})$ is a Riemannian manifold endowed with a subbundle $\mathrm{\Sigma }\subset \mathrm{End}\left(T\overline{M}\right)$ with a local basis $\left\{J_1,J_2,J_3\right\}$ satisfying

$$\overline{g}(J_{\alpha }X,J_{\alpha }Y)=\overline{g}(X,Y),J_{\alpha }^2=-I,J_{\alpha }{J}_{\alpha +1}=-J_{\alpha +1}{J}_{\alpha }=J_{\alpha +2},\alpha =1,2,3,$$

for vectors $X,Y\in T\overline{M}.$ If the bundle Σ is parallel with respect to the Levi–Civita connection of $\overline{g}$, then $(\overline{M},\mathrm{\Sigma },\overline{g})$ is said to be a quaternionic Kähler manifold. Clearly, such a manifold is of dimension $4m$ with $m\ge 1$. For a quaternionic Kähler manifold $(\overline{M},\mathrm{\Sigma },\overline{g})$ and a vector $0\ne X\in T\overline{M}$, the four-plane $\overline{Q}\left(X\right)$ spanned by $\{X,J_{1}X,J_{2}X,J_{3}X\}$ is called a quaternionic four-plane. Any two-plane in $\overline{Q}\left(X\right)$ is called a quaternionic plane. The sectional curvature of a quaternionic plane is called a quaternionic sectional curvature. A quaternionic Kähler manifold is called a quaternionic space form if it has a constant quaternionic sectional curvature.

We denote by $\overline{M}\left(c\right)$ a quaternionic space form of a constant quaternionic sectional curvature c. The curvature tensor $\overline{R}$ of $\overline{M}\left(c\right)$ satisfies

$$\begin{array}{cc}\hfill \overline{R}(X,Y)Z=& \frac{c}{4}\{\overline{g}(Z,Y)X-\overline{g}(X,Z)Y\hfill \\ & +\sum _{\alpha =1}^3[\overline{g}(Z,J_{\alpha }Y)J_{\alpha }X-\overline{g}(Z,J_{\alpha }X)J_{\alpha }Y+2\overline{g}(X,J_{\alpha }Y)J_{\alpha }Z]\}.\hfill \end{array}$$

In 2004, D. W. Yoon [132] extended Theorems 25 and 26 to the following.

Theorem 63.

Let M be an n-dimensional submanifold of a quaternionic projective $4m$-space $Q{P}^m\left(4c\right),c>0$. Then, we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n^2+8n-2)c.\end{array}$$

(47)

Theorem 64.

Let $M^n$ be an n-dimensional submanifold of $Q{H}^m\left(4c\right),c<0$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}(n+1)(n-2)c.\end{array}$$

(48)

A submanifold M of a quaternionic Kähler manifold $\overline{M}$ is said to be slant if, for each vector $0\ne X\in T_{p}M$, the angle $\theta \left(X\right)$ between $J_{\alpha }X$ and $T_{p}M$, $\alpha =1,2,3$ is a global constant. A slant submanifold of $\overline{M}$ is called proper if its slant angle $\theta \ne 0,\frac{\pi }{2}$.

For slant submanifolds of a quaternionic space form, G.-E. Vîlcu proved the following.

Theorem 65

([133,134]). Let M be an n-dimensional θ-slant submanifold of a quaternionic space form ${\overline{M}}^{4m}\left(c\right)$. Then, we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n-2}{2}\left\{\frac{n^2}{n-1}{\parallel H\parallel }^2+\frac{c}{4}(n+1+9{cos}^2\theta )\right\}.\end{array}$$

(49)

The equality in (49) holds at a point $p\in M$ if, and only if, there exist an orthonormal frame $\{e_1,\dots ,e_n\}$ of $T_{p}M$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{4m}\}$ of $T_p^{\perp }M$, such that the shape operators of M at p take the form (22).

Remark 28.

In [135], K. Dekimpe, J. Van der Veken and L. Vrancken constructed an explicit map from a generic minimal $\delta \left(2\right)$-ideal Lagrangian submanifold of ${\mathbb{C}}^n$ to the quaternionic projective space $Q{P}^{n-1}$, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for $n=3$. They also showed that there is a one-to-one correspondence between such surfaces in $Q{P}^2$ and minimal Lagrangian surfaces in $C{P}^2$.

Remark 29.

S. Decu-Marinescu derived in [136] the first Chen inequality for submanifolds in quaternion space forms endowed with a semisymmetric nonmetric connection. On the other hand, M. A. Lone derived in [137] the first Chen inequality for submanifolds in quaternion space forms endowed with a quarter-symmetric nonmetric connection.

21. Basics on Almost Contact Metric, Sasakian and Kenmotsu Manifolds

An odd-dimensional Riemannian manifold $(\overline{M},g)$ is called an almost contact metric manifold [138] if there exist a $(1,1)$-tensor field ϕ, a vector field ξ and a one-form η, such that

$$\begin{array}{cc}& \eta \left(\xi \right)=1,{\varphi }^2\left(X\right)=-X+\eta \left(X\right)\xi ,\varphi \xi =0,\eta \circ \varphi =0,\hfill \\ & \eta \left(X\right)=g(X,\xi ),g(\varphi X,\varphi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)\hfill \end{array}$$

for $X,Y\in \mathfrak{X}\left(\overline{M}\right)$. A submanifold M of an almost contact metric manifold $\overline{M}$ is called an anti-invariant if $\varphi \left(T_{p}M\right)\subset T_p^{\perp }M$ holds for every point $p\in M$. An almost contact metric manifold is called a contact metric manifold if it satisfies $g(X,\varphi Y)=d\eta (X,Y)$. An almost contact metric manifold $\overline{M}$ is called normal if the tensor field

$$N_{\varphi }=[\varphi ,\varphi ]+2d\eta \otimes \xi$$

vanishes identically, where $[\varphi ,\varphi ]$ is called the Nijenhuis tensor of ϕ. A normal contact metric manifold is said to be a Sasakian manifold. An almost contact metric manifold is Sasakian if, and only if, the Levi–Civita connection ∇ satisfies $\left({\nabla }_X\varphi \right)Y=g(X,Y)-\eta \left(Y\right)X$. A tangent plane π orthogonal to ${\xi }_p$ in $T_p\overline{M},p\in \overline{M},$ which is invariant under ϕ, is called a ϕ-section.

An almost contact metric manifold is called almost cosymplectic if it satisfies $d\eta =0$ and $d\varphi =0$ applied to a $(1,1)$-tensor. An almost cosymplectic manifold is called cosymplectic if it satisfies $\nabla \varphi =0$ and $\nabla \xi =0.$ An almost contact metric manifold $\overline{M}$ is called a Kenmotsu manifold if it satisfies

$$\left({\nabla }_X\varphi \right)Y=g(\varphi X,Y)\xi -\eta \left(Y\right)\varphi X.$$

A Sasakian space form is a Sasakian manifold with a constant ϕ-sectional curvature. The Riemann curvature tensor of a Sasakian space form $\overline{M}\left(c\right)$, with a constant ϕ-sectional curvature c, satisfies

$$\begin{array}{cc}\hfill \overline{R}(X,Y)Z& =\frac{c+3}{4}\left[g(Y,Z)X-g(X,Z)Y\right]\hfill \\ & +\frac{c-1}{4}\left[g(X,\varphi Z)\varphi Y-g(Y,\varphi Z)\varphi X+2g(X,\varphi Y)\varphi Z\right]\hfill \\ \hfill & +\frac{c-1}{4}\left[\eta \left(X\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)X+g(X,Z)\eta \left(Y\right)\xi -g(Y,Z)\eta \left(X\right)\xi \right].\hfill \end{array}$$

In [139], P. Alegre, D. E. Blair and A. Carriazo defined generalized Sasakian space forms; namely, an odd-dimensional manifold $\overline{M}$ equipped with an almost contact metric structure $(\varphi ,\xi ,\eta ,g)$ is called a generalized Sasakian space form if there are three functions $f_1$, $f_2$, $f_3$ on $\overline{M}$, such that the Riemann curvature tensor of $\overline{M}$ satisfies

$$\begin{array}{cc}\hfill \overline{R}(X,Y)Z=& f_1\left[g(Y,Z)X-g(X,Z)Y\right]\hfill \\ & +f_2\left[g(X,\varphi Z)\varphi Y-g(Y,\varphi Z)\varphi X+2g(X,\varphi Y)\varphi Z\right]\hfill \\ & +f_3\left[\eta \left(X\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)X+g(X,Z)\eta \left(Y\right)\xi -g(Y,Z)\eta \left(X\right)\xi \right].\hfill \end{array}$$

We simply denote such a manifold by $\overline{M}(f_1,f_2,f_3)$. A generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ is a Sasakian space form $\overline{M}\left(c\right)$ if $f_1=\frac{c+3}{4}$ and $f_2=f_3=\frac{c-1}{4}$ for a constant c.

An n-dimensional submanifold M of an almost contact metric manifold ${\overline{M}}^{2m+1}$ is said to be a C-totally real submanifold if the structure vector field ξ is normal to M. It is known that one has $\varphi \left(T_{p}M\right)\subset T_p^{\perp }M$ for C-totally real submanifolds. For any vector X tangent to a submanifold M of an almost contact metric manifold, we put

$$\begin{array}{c}\hfill \varphi X=PX+FX,\end{array}$$

(50)

where $PX$ and $FX$ denote the tangential and normal components of $\varphi X$, respectively.

22. Inequalities for Submanifolds in Sasakian Space Forms

In this section, we present many versions of first Chen inequalities for various submanifolds in Sasakian space forms.

22.1. Submanifolds Tangent to the Structure Vector Field in Sasakian Space Forms

The following result is due to Y. H. Kim and D.-S. Kim.

Theorem 66

([140,141]). Let M be an n-dimensional submanifold in a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$ with $c\ne 1$ whose structure vector field ξ is tangent to M. Then, we have

$$\begin{array}{cc}& \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{8}\left\{n(n-3)c+3n^2-n-8\right\}(\mathrm{if}c<1)\mathrm{or}\hfill \\ & \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{n+1}{8}\left\{(n-1)c+3n-7\right\}(\mathrm{if}c>1).\hfill \end{array}$$

Y. H. Kim and D.-S. Kim [141] also obtained a necessary and sufficient condition for either inequality to become an equality. Another version of the first Chen inequality for submanifolds tangent to the structure vector field ξ of a Sasakian space form was also derived by A. Carriazo in [142]. The following results are consequences of [142].

Proposition 3

([143]). If M is an n-dimensional invariant submanifold in a Sasakian space form $\overline{M}\left(c\right)$, then

$${\delta }^{\prime }\left(2\right)\le \frac{(c+3)(n-2)(n+1)}{8}+\frac{(c-1)(n-7)}{8},$$

where ${\delta }^{\prime }\left(2\right)=\tau -inf\left\{K\left(\pi \right)\right|\pi \subset D,\varphi \left(\pi \right)\subset \pi \}$.

Proposition 4

([143]). If M is an n-dimensional anti-invariant submanifold in a Sasakian space form $\overline{M}\left(c\right)$, then

$$\delta \left(2\right)\le \frac{n-2}{2}\left[\frac{n^2}{n-1}{∥H∥}^2+\frac{(c+3)(n+1)}{4}\right]-\frac{(c-1)(n-1)}{4}.$$

Remark 30.

For further results in this respect, see also [144].

22.2. Inequalities for C-Totally Real Submanifolds of Sasakian Space Forms

An n-dimensional submanifold M of an almost contact metric manifold ${\overline{M}}^{2m+1}$ is said to be a C-totally real submanifold if the structure vector field ξ is normal to M.

For C-totally real submanifolds, F. Defever, I. Mihai and L. Verstraelen proved the following.

Theorem 67

([145]). Let M be an n-dimensional C-totally real submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$. Then, we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{(n-2)(n+1)(c+3)}{8}.\end{array}$$

(51)

If the equality case of (51) holds identically, then M is minimal.

In [145], they also proved the following.

Theorem 68

([145]). Let M be an n-dimensional C-totally real submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$. If M has a constant scalar curvature, then M satisfies the equality in (51) identically if, and only if, either (i) M is a totally geodesic submanifold or (ii) $n=3,\overline{M}\left(c\right)=S^7\left(1\right)$, and M is locally congruent to an isometric immersion into $S^7\left(1\right)$ of a three-sphere $S^3$ with a special metric $g^{\prime }$.

Remark 31.

In [145], F. Defever, I.Mihai and L. Verstraelen also constructed a three-dimensional minimal C-totally real submanifold of Sasakian $S^7\left(1\right)$ satisfying inequality (51), but which is neither totally geodesic nor locally isometric to $(S^3,g^{\prime })$.

F. Defever, I. Mihai and L. Verstraelen studied C-totally real submanifolds of Sasakian space forms. They proved the following.

Theorem 69

([146]). Let M be an n-dimensional C-totally real submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$. If M satisfies the equality in (51), then, for all $X\in TM$, $\varphi \left(X\right)$ is perpendicular to the mean curvature vector H.

J. Bolton, F. Dillen, J. Fastenakels and L. Vrancken improved inequality (51) to the following.

Theorem 70

([147]). Let M be an n-dimensional C-totally real submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$. Then,

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n^2(2n-3)}{2(2n+3)}{\parallel H\parallel }^2+\frac{(n-2)(n+1)}{2}c.\end{array}$$

(52)

Moreover, if $n=m\ge 4$ and the equality is attained at every point, then M must be minimal.

The next result is due to I. Mihai.

Theorem 71

([148]). Every minimal C-totally real submanifold of a Sasakian space form $\overline{M}\left(c\right)$ with $c<-3$ is irreducible.

22.3. Inequalities for Legendrian Submanifolds of Sasakian Space Forms

A C-totally real submanifold of ${\overline{M}}^{2m+1}$ is called a Legendrian submanifold if $n=m$. Thus, a Legendrian submanifold is C-totally real with the smallest possible codimension.

The next result was due to F. Defever, I. Mihai and L. Verstraelen.

Theorem 72

([146]). Let M be a Legendrian submanifold of a Sasakian space form ${\overline{M}}^{2n+1}\left(c\right)$ with a nonconstant scalar curvature, such that the distribution $\mathcal{E}$ defines a differentiable subbundle and its complementary orthogonal subbundle ${\mathcal{E}}^{\perp }$ is involutive. Then, M satisfies

$$\begin{array}{c}\hfill \delta \left(2\right)=\frac{(n-2)(n+1)(c+3)}{8}\end{array}$$

if, and only if, M is locally congruent to the immersion

$$\begin{array}{c}\hfill \psi :\left(0,\frac{\pi }{2}\right){\times }_{cost}N{\times }_{sint}{S}^{n-3}\left(1\right),\psi (t,p,q)=(cost)p+(sint)q,\end{array}$$

where N is a C-totally real minimal surface of $S^5\left(1\right).$

22.4. Inequalities for CR-Submanifolds of Sasakian Space Forms

The odd dimensional analogue of CR-submanifolds in Kähler manifolds is the concept of contact CR-submanifolds in Sasakian manifolds; namely, a submanifold M in the Sasakian manifold $(\overline{M},\varphi ,\xi ,\eta ,g)$ carrying a ϕ-invariant distribution $\mathcal{D}$, i.e., $\varphi \left({\mathcal{D}}_p\right)\subset {\mathcal{D}}_p$ for any $p\in M$, such that the orthogonal complement ${\mathcal{D}}^{\perp }$ of $\mathcal{D}$ in $TM$ is ϕ anti-invariant, i.e., $\varphi \left({\mathcal{D}}_p^{\perp }\right)\subset {\mathcal{D}}_p^{\perp }$ for any $p\in M$. The contact CR-submanifold is called proper if both distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ are nontrivial distributions.

In [149], M. I. Munteanu and L. Vrancken studied $\delta \left(2\right)$-ideal contact CR-submanifolds and proved the following.

Theorem 73

([149]). Let $M^n$ be a proper, minimal $\delta \left(2\right)$-ideal contact CR-submanifold of Sasakian $S^{2m+1}\left(1\right)$. Then, n is even and there exists a totally geodesic Sasakian $S^{n+1}\left(1\right)$ in $S^{2m+1}\left(1\right)$ containing $M^n$ as a hypersurface.

Theorem 74

([149]). Let $M^n(n$ even) be a proper, minimal $\delta \left(2\right)$-ideal contact CR-submanifold of $S^{n+1}$. Then, M is locally the unit normal bundle of the Clifford torus $S^1(1/\sqrt{2})\times S^1(1/\sqrt{2})$ in $S^3\left(1\right)\subset S^{n+1}\left(1\right)$.

22.5. Inequalities for Slant Submanifolds of Sasakian Space Forms

For slant submanifolds of Sasakian space forms, A. Mihai proved the following.

Theorem 75

([99]). Let M be a $(2k+1)$-dimensional contact θ-slant submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$. Then, we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{(n-2)}{2}\left[\frac{n^2}{n-1}{\parallel H\parallel }^2+\frac{(n+1)(c+3)}{4}\right]+\frac{c-1}{8}[3(n-3){cos}^2\theta -2(n-1)].\end{array}$$

(53)

The equality in (53) holds identically if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal frame $\{e_1,\dots ,e_n=\xi \}$ of $T_{p}M$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m},e_{2m+1}\}$ of $T_p^{\perp }M$.

Remark 32.

A special class of a contact θ-slant submanifold of a Sasakian space form ${\overline{M}}^{2m+1}\left(c\right)$ was obtained by A. Mihai and D. Cioroboiu in [143].

22.6. Some Further Results

Remark 33.

A. Mihai and I. N. Rǎdulescu provided the Sasakian version of Theorem 32 in [150] for invariant submanifolds of Sasakian space forms.

Remark 34.

For first Chen inequalities for submanifolds of generalized Sasakian space forms endowed with a semisymmetric and quarter-symmetric connection, see [151,152,153].

Remark 35.

For further results on the first Chen inequality for submanifolds of generalized Sasakian space forms and of generalized complex space forms, see [154].

23. Inequalities for Submanifolds in Kenmotsu Space Forms

In 1969, S. Tanno [155] classified connected almost contact Riemannian manifolds whose automorphism group have the maximum dimensions into three classes; namely:

1.

Homogeneous normal contact Riemannian manifolds with a constant ϕ-holomorphic sectional curvature;

2.

Global Riemannian products of a line or a circle and a Kählerian space form;

3.

Warped product manifolds $L{\times }_{f}F$, where L is a line, F is a Kähler manifold and f is a differentiable function on F.

In 1972, K. Kenmotsu [156] studied the third class and characterized it by a tensor equation. Later, such a manifold was called a Kenmotsu manifold. In fact, a Kenmotsu space form $\overline{M}\left(c\right)$ of a constant ϕ-sectional curvature c is a generalized Sasakian space form with $f_1=\frac{c-3}{4}$ and $f_2=f_3=\frac{c+1}{4}$.

In [157], K. Matsumoto, I. Mihai and M. H. Shahid defined a generalized contact CR-submanifold of a Kenmoutsu manifold $(\overline{M},\varphi ,\xi ,\eta ,g)$ as a submanifold M tangent to ξ in $\overline{M}$, such that the maximal anti-invariant subspaces by ϕ: ${\mathcal{D}}_x^{\perp }=T_{x}M\cap \varphi \left(T_x^{\perp }M\right),x\in M$ defines a differentiable subbundle of $TM$.

In [157], they studied contact CR-submanifolds of a Kenmotsu space form. In particular, they proved the following.

Theorem 76

([157]). Let M be an $(n+1)$-dimensional generalized contact CR-submanifold in a Kenmotsu space form $\overline{M}\left(c\right)$ tangent to ξ. Then, we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{n-1}{2}\left[\frac{{(n+1)}^2}{n}{\parallel H\parallel }^2+\frac{1}{4}\left\{(n+2)(c-3)-\frac{2(n-3\ell )(c+1)}{n-1}\right\}\right],\ell =\frac{1}{2}dim\mathcal{D}.\end{array}$$

In [158], M. Mohammed investigated bi-slant submanifolds in Kenmotsu space forms. He obtained the next result.

Theorem 77.

Let M be an n-dimensional proper bi-slant submanifold in a Kenmotsu space form ${\overline{M}}^{2m+1}\left(c\right)$. Then, the following applies:

(1) 

For any plane section π-invariant by P (defined by (50)) and tangent to ${\mathcal{D}}_1$, we have:

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{n(n-3)(c-3)}{8}-(n-1)\hfill \\ & +\frac{3(c+1)}{4}\left[(d_1-1){cos}^2{\theta }_1+d_2{cos}^2{\theta }_2\right].\hfill \end{array}$$

(54)

(2) 

For any plane section π-invariant by P and tangent to ${\mathcal{D}}_2$, we have:

$$\begin{array}{cc}\hfill \tau -K\left(\pi \right)\le & \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{n(n-3)(c-3)}{8}-(n-1)\hfill \\ & +\frac{3(c+1)}{4}\left[d_1{cos}^2{\theta }_1+(d_2-1){cos}^2{\theta }_2\right],\end{array}$$

(55)

where we follow the same notations in Section 13 for bi-slant submanifolds.

The equality in (54) (or (55)) holds at a point $p\in M$ if, and only if, the shape operators of M at p take the form (22), with respect to some suitable orthonormal frame $\{e_1,\dots ,e_n=\xi \}$ of $T_{p}M$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m},e_{2m+1}\}$ of $T_p^{\perp }M$.

Remark 36.

For first Chen-type inequalities for bi-slant submanifolds in Kenmotsu space forms, see also [159,160] by P. K. Pandey, R. S. Gupta, I. Ahmad and A. Sharfuddin.

24. Submanifolds of Cosymplectic Space Forms

A cosymplectic space form $\overline{M}(f_1,f_2,f_3)\left(c\right)$ of a constant ϕ-sectional curvature c is a generalized Sasakian space form with $f_1=f_2=f_3=\frac{c}{4}$. For submanifolds of a cosymplectic space form, J.-S. Kim and J. Choi proved the following.

Theorem 78

([161]). Let M be an $(n+1)$-dimensional submanifold in a cosymplectic space form ${\overline{M}}^{2m+1}\left(c\right)$, such that the structure vector field ξ is tangent to M. Then, for each point $p\in M$ and each plane section $\pi \subset T_{p}M$, we have

$$\begin{array}{c}\hfill \tau -K\left(\pi \right)\le \frac{{(n+1)}^2(n-1)}{2n}{\parallel H\parallel }^2+\frac{c}{8}\left({3\parallel P\parallel }^2-6\alpha \left(\pi \right)+2\beta \left(\pi \right)+(n+1)(n-2)\right),\end{array}$$

(56)

where $\alpha \left(\pi \right)={〈e_1,P{e}_2〉}^2$ and $\beta \left(\pi \right)={\left(\eta \left(e_1\right)\right)}^2+{\left(\eta \left(e_2\right)\right)}^2$ with $\pi =\mathrm{Span}\{e_1,e_2\}$.

The equality in (56) holds at a point $p\in M$ if, and only if, there exist an orthonormal frame $\{e_1,\dots ,e_{n+1}\}$ of $T_{p}M$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m},e_{2m+1}\}$ of $T_p^{\perp }M$, such that (a) $\pi =\mathrm{Span}\{e_1,e_2\}$ and (b) the shape operators of M at p take the form (22).

A submanifold M of an almost contact metric manifold $\overline{M}$ with $\xi \in TM$ is called a semi-invariant submanifold of $\overline{M}$ if

$$\begin{array}{c}\hfill TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp }\oplus \xi ,\end{array}$$

(57)

where $\mathcal{D}=TM\cap \varphi \left(TM\right)$ and ${\mathcal{D}}^{\perp }=TM\cap \varphi \left(T^{\perp }M\right)$. In fact, condition (57) implies that $\mathrm{rank}\left(P\right)=dim\left(\mathcal{D}\right)$. A semi-invariant submanifold M becomes an invariant or anti-invariant submanifold accordingly, as the anti-invariant distribution ${\mathcal{D}}^{\perp }$ is $\left\{0\right\}$ or the invariant distribution $\mathcal{D}$ is $\left\{0\right\}$.

Theorem 79

([161]). Let M be an $(n+1)$-dimensional submanifold in a cosymplectic space form ${\overline{M}}^{2m+1}\left(c\right)$, such that the structure vector field ξ is tangent to M. If $c<0$, then we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{{(n+1)}^2(n-1)}{2n}{\parallel H\parallel }^2+\frac{c}{8}(n+1)(n-2).\end{array}$$

(58)

The equality in (58) holds if, and only if, M is a semi-invariant submanifold with $dim\mathcal{D}=2$.

Theorem 80

([161]). Let M be an $(n+1)$-dimensional submanifold in a cosymplectic space form ${\overline{M}}^{2m+1}\left(c\right)$, such that the structure vector field ξ is tangent to M. If $c>0$, then we have

$$\begin{array}{c}\hfill \delta \left(2\right)\le \frac{{(n+1)}^2(n-1)}{2n}{\parallel H\parallel }^2+\frac{c}{8}n(n+2).\end{array}$$

(59)

The equality in (59) holds if, and only if, M is an invariant totally geodesic cosymplectic space form.

Remark 37.

In [162], C. Özgür and C. Murathan derived the first Chen inequality for submanifolds of a cosymplectic space form endowed with a semisymmetric metric connection. And, in [163], they derived the corresponding inequality for submanifolds of a locally conformal almost cosymplectic manifold endowed with a semisymmetric metric connection.

Remark 38.

For first Chen-type inequalities for bi-slant submanifolds in cosymplectic space forms, see also R. S. Gupta’s article [164].

25. Inequalities for Submanifolds in $(\mathit{\kappa },\mathit{\mu })$-Contact Space Forms

A contact metric manifold $\left(\overline{M},\varphi ,\xi ,\eta ,g\right)$ is called a generalized $\left(\kappa ,\mu \right)$-space if its curvature tensor satisfies

$$\overline{R}\left(X,Y\right)\xi =\kappa [\eta \left(Y\right)X-\eta \left(X\right)Y]+\mu [\eta \left(Y\right)hX-\eta \left(X\right)hY]$$

for some functions $\kappa ,\mu$, where $h=\frac{1}{2}{\mathcal{L}}_{\xi }\varphi$ and $\mathcal{L}$ is the Lie derivative. If $\kappa ,\mu$ are constant, then $\overline{M}$ is called a $(\kappa ,\mu )$-space. A $(\kappa ,\mu )$-space is called a $(\kappa ,\mu )$-contact space form if it has a constant ϕ-sectional curvature c, which is denoted by $\tilde{M}\left(c\right)$.

In 2002, K. Arslan, R. Ezentas, I. Mihai and C. Murathan studied submanifolds of $(\kappa ,\mu )$-contact space forms. They obtained the following.

Theorem 81

([165]). Let M be a submanifold in a $(\kappa ,\mu )$-contact space form ${\tilde{M}}^{2m+1}\left(c\right)$, such that the structure vector field ξ is normal to M. Then, we have:

(1) 

For any invariant plane section $\pi \subset T_{p}M$,

$$\begin{array}{c}\hfill \tau -K\left(\pi \right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{n}{4}(1-\kappa )(n-1)+\frac{3}{4}(\kappa +1){\parallel P\parallel }^2+(1+2\kappa ).\end{array}$$

(60)

(2) 

For any anti-invariant plane section $\pi \subset T_{p}M$,

$$\begin{array}{c}\hfill \tau -K\left(\pi \right)\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{n}{4}(1-\kappa )(n-1)+\frac{3}{4}(\kappa +1){\parallel P\parallel }^2-\frac{1}{2}(1-\kappa ).\end{array}$$

(61)

The equality in (60) and (61) holds at a point $p\in M$ if, and only if, there exist an orthonormal frame $\{e_1,\dots ,e_n\}$ of $T_{p}M$ and an orthonormal normal frame $\{e_{n+1},\dots ,e_{2m},e_{2m+1}=\xi \}$ of $T_p^{\perp }M$, such that the shape operators of M at p take the form (22).

Remark 39.

The first Chen inequality of submanifolds M of a $(\kappa ,\mu )$-contact space form, such that the structure vector field ξ is tangent to M, was derived by M. M. Tripathi in [166].

Remark 40.

The corresponding first Chen inequality for C-totally real submanifolds in $(\kappa ,\mu )$-contact space forms ${\tilde{M}}^{2m+1}\left(c\right)$ was obtained by M. M. Tripathi and J.-S. Kim in [167].

Remark 41.

The corresponding first Chen inequalities for submanifolds in a $(\kappa ,\mu )$-contact space form endowed with a semisymmetric metric or nonmetric connection were derived in [168] and [169], respectively. Further, the inequality for submanifolds in $(\kappa ,\mu )$-contact space forms endowed with a generalized semisymmetric, nonmetric connection was obtained by Y. Wang in [170].

Remark 42.

The notion of generalized $(\kappa ,\mu )$-space forms was defined in [171]. The first Chen-type inequalities for invariant and C-totally real submanifolds in generalized $(\kappa ,\mu )$-space forms were derived in [172,173], respectively.

26. Inequality for Submanifolds in $\mathit{S}$- and $\mathit{T}$-Space Forms

The notion of an f-structure was defined by K. Yano in [174]; namely, a tensor field f of the type $(1,1)$ of the rank $2m$ on a Riemannian manifold $(M,g)$ is called an f-structure if it satisfies $f^3+f=0$. A Riemannian $(2m+s)$-manifold $(M,g)$ equipped with an f-structure is called a metric f-manifold if there exist s global vector fields ${\xi }_1,\cdots ,{\xi }_s$ on M, such that

$$\begin{array}{c}f{\xi }_{\alpha }=0,{\eta }_{\alpha }\circ f=0,f^2=-I+\sum _{\alpha =1}^s{\eta }_{\alpha }\otimes {\xi }_{\alpha },\hfill \\ g(X,Y)=g(fX,fY)+\sum _{\alpha =1}^s{\eta }_{\alpha }\left(X\right){\eta }_{\alpha }\left(Y\right)\end{array}$$

(62)

for $X,Y\in \mathfrak{X}\left(M\right)$, where ${\eta }_1,\dots ,{\eta }_s$ are the dual one-forms of ${\xi }_1,\dots ,{\xi }_s$. Let F be the fundamental $two$-form on M defined by $F(X,Y)=g(X,fY)$. Then, a metric f-manifold is called normal if the Nijenhuis tensor $\left[f,f\right]$ of f satisfies

$$[f,f]=-2\sum _{\alpha =1}^s{\xi }_{\alpha }\otimes d{\eta }_{\alpha }.$$

A normal metric f-manifold is called an S-manifold if $F=d{\eta }_{\alpha }$ for $\alpha \in \left\{1,\dots ,s\right\}$.

For a T-manifold, beside the relations in (62), the following conditions also hold:

$$\begin{array}{cc}& \left({\nabla }_X\varphi \right)Y=0{\nabla }_X{\xi }_i=0,\hfill \end{array}$$

(63)

for any vector fields $X,Y$ tangent to M.

A plane section on a metric f-manifold M is called an f-section if it is spanned by a unit vector X orthogonal to all structure vector fields ${\xi }_1,\dots ,{\xi }_s$ and $fX$. The sectional curvature of an f-section is called an f-sectional curvature. A T-manifold is called a T-space form if it has a constant f-sectional curvature c, denoted by $\widehat{M}\left(c\right)$.

The corresponding first Chen inequalities of submanifolds in S and T-space forms were studied in [175,176].

27. Inequalities for Riemannian Maps

As a generalization of isometric immersions and Riemannian submersions, A. E. Fischer [177] introduced, in 1992, the notion of Riemannian maps between Riemannian manifold as follows:

Let $F:(M,g_M)\to (N,g_N)$ be a map between two Riemannian manifolds, such that $0<\mathrm{rank}F<min\{n,m\}$ with $n=dimM$ and $m=dimN$. Let $ker{F}_{*}$ denote the kernel space of $F_{*}$ and $\mathcal{H}={(ker{F}_{*})}^{\perp }$ the orthogonal complementary space of $ker{F}_{*}$. So, we have the direct decomposition:

$$TM=ker{F}_{*}\oplus \mathcal{H}.$$

Let $\mathrm{range}{F}_{*}$ be the range of $F_{*}$ and ${\left(\mathrm{range}{F}_{*}\right)}^{\perp }$ the orthogonal complementary space of $\mathrm{range}{F}_{*}$ in $TN$. Since $\mathrm{rank}F<min\{n,m\}$, we have ${\left(\mathrm{range}{F}_{*}\right)}^{\perp }\ne \left\{0\right\}$. Hence, $TN$ has the decomposition:

$$TN=\left(\mathrm{range}{F}_{*}\right)\oplus {\left(\mathrm{range}{F}_{*}\right)}^{\perp }.$$

A smooth map $F:(M,g_M)\to (N,g_N)$ is called a Riemannian map at $p\in M$ if the horizontal restriction $F_{{*}_p}^h:{(ker{F}_{*p})}^{\perp }\to \left(\mathrm{range}{F}_{*p}\right)$ is a linear isometry between the inner product spaces $({(ker{F}_{*p})}^{\perp },g_M\left(p\right){|}_{{(ker{F}_{*p})}^{\perp }})$ and $(\mathrm{range}{F}_{*p},g_N\left(q\right){|}_{\left(\mathrm{range}{F}_{*p}\right)})$ with $q=F\left(p\right)$. For a Riemannian map $F:(M,g_M)\to (N,g_N)$, the second fundamental form of F is defined by

$$(\nabla F_{*})(X,Y)={\nabla }_X^F{F}_{*}\left(Y\right)-F_{*}\left({\nabla }_X^{M}Y\right)$$

for $X,Y\in \mathfrak{X}\left(M\right)$, where ∇ is the induced connection from the Levi–Civita connection ${\nabla }^M$ on M. Let $R^M$ and $R^N$ be the curvature tensors for $(M,g_M)$ and $(N,g_N)$, respectively.

It is easy to see that isometric immersions and Riemannian submersions are Riemannian maps with $ker{F}_{*}=\left\{0\right\}$ and ${\left(\mathrm{range}{F}_{*}\right)}^{\perp }=\left\{0\right\}$, respectively.

In 2016, B. Şahin [178] established the first Chen inequality for Riemannian maps from Riemannian manifolds to a real space form as follows (see also [179]).

Theorem 82

([178]). Let $F:(M,g_M)\to (R\left(c\right),\overline{g})$ be a Riemannian map from a Riemannian manifold $(M,g_M)$ to a real space form $(R\left(c\right),\overline{g})$ with the $\mathrm{rank}F=r\ge 3$. Then, for each point $p\in M$ and each plane section $p\in T_{p}M$, we have

$$\begin{array}{c}\hfill {\rho }_{\mathcal{H}}-K\left(\pi \right)\le \frac{r-2}{2}\left\{\frac{1}{r-1}{\parallel {\tau }^{\mathcal{H}}\parallel }^2+(r+1)c\right\},\end{array}$$

(64)

where ${\rho }_{\mathcal{H}}$ is the scalar curvature defined by $\mathcal{H}={(ker{F}_{*})}^{\perp }$, and ${\tau }^{\mathcal{H}}$ is defined by

$${\tau }^{\mathcal{H}}=\sum _{i=1}^r\overline{g}(\nabla F_{*})(e_i,e_i),\nabla F_{*})(e_i,e_i).$$

The equality in (64) holds identically if, and only if, there exist an orthonormal basis $\{e_1,\dots ,e_r\}$ of ${(ker{F}_{*})}^{\perp }$ and an orthonormal basis $\{V_{r+1},\dots ,V_{r+d}\}$ of ${\left(\mathrm{range}{F}_{*}\right)}^{\perp }$, such that the shape operator takes the form:

$$\begin{array}{cc}\hfill & S_{r+1}=\left(\begin{array}{ccccc}a& 0& 0& \dots & 0\\ 0& \mu -a& 0& \dots & 0\\ 0& 0& \mu & \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & \mu \end{array}\right),S_{\beta }=\left(\begin{array}{ccccc}{h}_{11}^r& h_{12}^r& 0& \dots & 0\\ h_{12}^r& -h_{11}^r& 0& \dots & 0\\ 0& 0& 0& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 0\end{array}\right),\beta =r+2,\dots ,r+d.\hfill \end{array}$$

Remark 43.

The method used by B. Şahin in [178] has been applied by several geometers to investigate either the first Chen inequality or Chen–Ricci inequalities for various types of Riemannian submersions in [180,181,182,183,184,185,186,187], among some others.

28. Statistical Manifolds and Statistical Submanifolds

A statistical structure can be regarded as a generalization of a Riemannian structure. The theory of abstract generalizations of statistical models as statistical manifolds is a fast growing area of research in differential geometry.

The notion of statistical manifolds was proposed, in 1985, by S. Amari [188], which brings a framework for the field of information geometry.

Let $(\widehat{M},\widehat{g})$ be a Riemannian manifold with a pair of torsion-free affine connections $\widehat{\nabla }$ and ${\widehat{\nabla }}^{*}$. Then, $(\widehat{\nabla },\widehat{g})$ is called a statistical structure on $\widehat{M}$ if

$$\begin{array}{c}\hfill \left({\widehat{\nabla }}_X\widehat{g}\right)(Y,Z)-\left({\widehat{\nabla }}_Y\widehat{g}\right)(X,Z)=0\end{array}$$

for any vector fields $X,Y,Z$ tangent to $\widehat{M}$. A Riemannian manifold $(\widehat{M},\widehat{g})$, with a statistical structure satisfying the compatibility condition

$$\begin{array}{c}\hfill X\widehat{g}(Y,Z)=\widehat{g}({\widehat{\nabla }}_{X}Y,Z)+\widehat{g}(Y,{\widehat{\nabla }}_X^{*}Z),\end{array}$$

is called a statistical manifold and is denoted as $(\widehat{M},\widehat{g},\widehat{\nabla },{\widehat{\nabla }}^{*})$. Any torsion-free connection $\widehat{\nabla }$ has a dual connection ${\widehat{\nabla }}^{*}$ and satisfies

$$2{\widehat{\nabla }}^{\circ }=\widehat{\nabla }+{\widehat{\nabla }}^{*},$$

where ${\widehat{\nabla }}^{\circ }$ is the Levi–Civita connection on $\widehat{M}$.

The curvature tensor fields, with respect to the dual connections $\widehat{\nabla }$ and ${\widehat{\nabla }}^{*}$, are denoted by $\widehat{R}$ and $\widehat{R^{*}}$. The curvature tensor field ${\widehat{R}}^{\circ }$ associated with ${\widehat{\nabla }}^{\circ }$ is called a Riemannian curvature tensor. Generically, the dual connections are not metric; one cannot define the sectional curvature’s statistical settings as in the case of Riemannian geometry. A notable difference here is that while writing the curvature (sectional), in addition to the contribution showed by the dual connection via $R^{*}$, there is a correction term in the form of $R^{\circ }$ contributed by the Levi–Civita connection ${\nabla }^{\circ }.$ In this connection, B. Opozda proposed two notions of sectional curvature on statistical manifolds (see [189,190]).

Let $\widehat{M}$ be a statistical manifold and π a plane section in $T\widehat{M}$, with orthonormal basis $\{X,Y\}$, then the sectional K-curvature is defined in [190] as

$$\begin{array}{c}\hfill \widehat{K}\left(\pi \right)=\frac{1}{2}\left[\widehat{g}(\widehat{R}(X,Y)Y,X)+\widehat{g}({\widehat{R}}^{*}(X,Y)Y,X)-\widehat{g}({\widehat{R}}^{\circ }(X,Y)Y,X)\right].\end{array}$$

The curvature tensors $\widehat{R}$ and $\widehat{R^{*}}$ satisfy the following property:

$$\begin{array}{c}\hfill \widehat{g}(\widehat{R}(X,Y)Y,W)=-\widehat{g}({\widehat{R}}^{*}(X,Y)W,Z).\end{array}$$

Let $(M,g,\nabla ,{\nabla }^{*})$ be an n-dimensional statistical submanifold of an m-dimensional statistical manifold $({\widehat{M}}^m,\widehat{g},\widehat{\nabla },{\widehat{\nabla }}^{*})$. Then, the Gauss and Weingarten formulas of M in $\widehat{M}$ are given, respectively (see, e.g., [191]):

$$\begin{array}{ccc}& & {\widehat{\nabla }}_{X}Y={\nabla }_{X}Y+\sigma (X,Y),{\widehat{\nabla }}_X\xi =-A_{\xi }X+{\nabla }_X^{\perp }\xi \hfill \\ & & {\widehat{\nabla }}_X^{*}Y={\nabla }_X^{*}Y+{\sigma }^{*}(X,Y),{\widehat{\nabla }}_X^{*}\xi =-A_{\xi }^{*}X+{\nabla }_X^{*\perp }\xi \hfill \end{array}$$

for all $X,Y\in TM$ and $\xi \in T^{\perp }M$. Moreover, we have the following equations:

$$\begin{array}{ccc}& & Xg(Y,Z)=g({\nabla }_{X}Y,Z)+g(Y,{\nabla }_X^{*}Z)\hfill \\ & & \widehat{g}(\sigma (X,Y),\xi )=g(A_{\xi }^{*}X,Y),\widehat{g}({\sigma }^{*}(X,Y),\xi )=g(A_{\xi }X,Y)\hfill \\ & & X\widehat{g}(\xi ,\eta )=\widehat{g}({\nabla }_X^{\perp }\xi ,\eta )+\widehat{g}(\xi ,{\nabla }_X^{*\perp }\eta ).\hfill \end{array}$$

The mean curvature vector fields for the orthonormal tangent and normal frames $\{e_1,e_2,\dots ,e_n\}$ and $\{e_{n+1},e_{n+2},\dots ,e_{4m}\}$ are defined as

$$H=\frac{1}{n}\sum _{i=1}^n\sigma (e_i,e_i)=\frac{1}{n}\sum _{\gamma =1}^m\left(\sum _{i=1}^n{\sigma }_{ii}^{\gamma }\right){\xi }_{\gamma },{\sigma }_{ij}^{\gamma }=g(\sigma (e_i,e_j),e_{\gamma })$$

and

$$H^{*}=\frac{1}{n}\sum _{i=1}^n{\sigma }^{*}(e_i,e_i)=\frac{1}{n}\sum _{\gamma =1}^m\left(\sum _{i=1}^n{\sigma }_{ii}^{*\gamma }\right){\xi }_{\gamma },{\sigma }_{ij}^{*\gamma }=g({\sigma }^{*}(e_i,e_j),e_{\gamma })$$

for $1\le i,j\le n$ and $1\le l\le m$.

29. Inequalities for Statistical Submanifolds in Statistical Manifolds

Now, we present first Chen-type inequalities for statistical submanifolds in various classes of statistical manifolds.

29.1. Inequality for Statistical Submanifolds of HESSIAN Space Forms

For a Hessian manifold $\tilde{M}$ with a Hessian structure $(\tilde{g},\tilde{\nabla })$, put $\gamma =\tilde{\nabla }-{\tilde{\nabla }}^{\circ }$. Then, the Hessian curvature of $(\tilde{g},\tilde{\nabla })$ is given by (see [192])

$$Q(\tilde{X},\tilde{Y}):=[{\gamma }_{\tilde{X}},{\gamma }_{\tilde{Y}}],\tilde{X},\tilde{Y}\in \mathfrak{X}\left(\tilde{M}\right).$$

Note that $Q=\frac{1}{2}(\tilde{R}+{\tilde{R}}^{*})-{\tilde{R}}^g.$

A corresponding notion of sectional curvature can be defined in Hessian manifolds as follows: Let π be a plane field and $\{\tilde{X},\tilde{Y}\}$ an orthonormal basis of π. H. Shima [192] defined Hessian sectional curvature as $\tilde{K}\left(\pi \right):=g(Q(\tilde{X},\tilde{Y})\tilde{Y},\tilde{X}).$ A statistical structure $(\tilde{g},\tilde{\nabla })$ is said to be of a constant Hessian curvature c if its Hessian sectional curvature satisfies

$$Q(\tilde{X},\tilde{Y})\tilde{Z}=\frac{c}{2}\left[\tilde{g}(\tilde{Y},\tilde{Z})\tilde{X}+g(\tilde{X},\tilde{Y})\tilde{Z}\right].$$

Note that a manifold of a constant Hessian curvature c is a space form of a constant sectional curvature $-\frac{c}{4}$.

The study of statistical submanifolds in Hessian manifolds was started by A. Mihai and I. Mihai in [193]. More precisely, they dealt with statistical submanifolds in Hessian manifolds of a constant Hessian curvature.

For statistical submanifolds in a Hessian manifold of a constant Hessian curvature, the first author, A. Mihai and I. Mihai [194] obtained the following.

Theorem 83.

Let M be an n-dimensional statistical submanifold in a Hessian manifold ${\tilde{M}}^m\left(c\right)$ of a constant Hessian curvature c. Then, for any $p\in M$ and any plane section π at p, we have:

$$\begin{array}{c}\hfill {\tau }_0-K_0\left(\pi \right)\le \tau -K\left(\pi \right)+\frac{n^2(n-2)}{2(n-1)}{(\parallel H\parallel }^2+\parallel H^{*}{\parallel }^2)-(n-2)(n+1)\frac{c}{4},\end{array}$$

(65)

where ${\tau }_0$ and $K_0$ are the scalar curvature and the sectional curvature of M with respect to the Riemann curvature tensor and τ and K, with respect to the Hessian curvature tensor.

An immediate consequence of Theorem 83 is the following.

Corollary 4

([194]). Let M be a statistical submanifold in a Hessian manifold ${\tilde{M}}^m\left(c\right)$ of a constant Hessian curvature c. If there exist a point $p\in M$ and a plane section π at p, such that

$$(\tau -K\left(\pi \right))-({\tau }_0-K_0\left(\pi \right))<(n-2)(n+1)\frac{c}{4},n=dimM,$$

then M is nonminimal in ${\tilde{M}}^m\left(c\right)$, i.e., either $H\ne 0$ or $H^{*}\ne 0$.

Remark 44.

I. Mihai and R.-I. Mihai [195] extended inequality (65) in Theorem 83 to a more general inequality involving mutually orthogonal k plane sections ${\pi }_i\subset T_{p}M,i=1,\dots ,k,$ at any point $p\in M$.

On the other hand, Theorem (83) was extended further by H. Furuhata, I. Hasegawa and N. Satoh in [196] to an inequality involving an arbitrary δ-invariant, $\delta (n_1,\dots ,n_k)$, for a statistical submanifold in a Hessian manifold ${\tilde{M}}^m\left(c\right)$ of a constant Hessian curvature.

29.2. Inequality for Statistical Submanifolds of Statistical Manifolds of a Quasi-Constant Curvature

Analogous to [68], a statistical structure $(g,\nabla )$ is said to be of a quasi-constant curvature $c\in \mathbb{R}$ (with respect to the Riemannian metric g) if its curvature tensor R is of the form:

$$\begin{array}{c}R(X,Y)Z=a\left[g(Y,Z)X-g(X,Z)Y\right]\hfill \\ +b\left[\omega \left(Y\right)\omega \left(Z\right)X-g(X,Z)\omega \left(Y\right)\zeta +g(Y,Z)\omega \left(X\right)\zeta -\omega \left(X\right)\omega \left(Y\right)\zeta \right],\hfill \end{array}$$

where a and b are smooth functions and ω is the dual one-form of a unit vector field ζ. In particular, a statistical structure $(g,\nabla )$ is said to be of a constant curvature c if

$$R(X,Y)Z=c\left[g(Y,Z)X-g(X,Z)Y\right]$$

for a constant c. When $c=0$, it is called a Hessian structure.

If $(g,\nabla )$ is a statistical structure of a quasi-constant curvature (in particular, of a constant curvature c), then $(g,{\nabla }^{*})$ is also a statistical structure of a quasi-constant curvature (in particular, of a constant curvature c). Further, if $(g,\nabla )$ is a Hessian structure, then $(g,{\nabla }^{*})$ is also a Hessian structure (see [197]).

In 2018, H. Aytimur and C. Özgür [198] studied statistical submanifolds of statistical manifolds of a quasi-constant curvature in which they obtained the Chen–Ricci inequality and generalized the Wingten inequality. In [199], P. Bansal, S. Uddin and M. H. Shahid derived a first Chen-type inequality for statistical submanifolds of a statistical manifold of a quasi-constant curvature. Also, G. He, J. Zhang and P. Zhao derived in [200] a first Chen-type inequality for a statistical submanifold of a statistical manifold M of a constant curvature, such that M admits a nonintegrable distribution on M with a constant rank.

29.3. Inequality for Statistical Submanifolds of Kähler-like Statistical Space Forms

Let $\widehat{M}$ be a differentiable manifold. Assume that there is a rank three-bundle Λ of $End\left(T\widehat{M}\right)$ and there exists a local basis $\left\{J_{\alpha }\right\}$ acting on the section of Λ satisfying

$$J_{\alpha }^2=-I,J_{\alpha }{J}_{\alpha +1}=-J_{\alpha +1}{J}_{\alpha }=J_{\alpha +2},$$

for $\alpha =1,2,3$, where I denotes the identity map. In such a case, $\left\{J_{\alpha }\right\}$ is called a canonical basis of Λ and Λ is called an almost quaternion structure on $\widehat{M}$. Moreover, ($\widehat{M},\mathrm{\Lambda }$) is called almost quaternionic with the dimension $4m,$ $m\ge 1$. A Riemannian metric g on $\widehat{M}$ is said to be adapted to the almost quaternionic structure Λ if it satisfies

$$g(J_{\alpha }X,J_{\alpha }Y)=g(X,Y),\alpha \in \{1,2,3\}$$

(66)

for all vector fields $X,Y$ on $\widehat{M}$ and any canonical basis $\{J_1,J_2,J_3\}$ of $\mathrm{\Lambda }.$ Let $(\widehat{M},g)$ be a Riemannian manifold with an almost quaternion structure Λ having $\{J_1,J_2,J_3\}$ as its canonical basis and three other $(1,1)$-tensor fields of type $\{J_1^{*},J_2^{*},J_3^{*}\}$ satisfying

$$g(J_{\alpha }X,Y)+g(X,J_{\alpha }^{*}Y)=0$$

(67)

for all vector fields $X,Y$ on $\widehat{M}.$ Then, $(\widehat{M},\mathrm{\Lambda },g)$ is called an almost Hermite-like quaternion manifold and, if this manifold is endowed with the torsion-free and symmetric connection pair $(\nabla ,{\nabla }^{*})$, then $(M,\widehat{\nabla },\mathrm{\Lambda },g)$, it is called an almost Hermite-like quaternion statistical manifold. If this $J^{*}$ satisfies (66), then we can consider a subbundle of $End\left(T\widehat{M}\right)$ locally spanned by $\{J_1^{*},J_2^{*},J_3^{*}\}$, such that ${\left(J_{\alpha }^{*}\right)}^{*}=J_{\alpha }$ and $g(J_{\alpha }X,J_{\alpha }^{*}Y)=g(X,Y)$ for vector fields $X,Y$ on $\mathfrak{X}\left(\widehat{M}\right)$ and $\alpha \in \{1,2,3\}$. If $(\widehat{M},\widehat{\nabla },\mathrm{\Lambda },g)$ is an almost Hermite-like quaternionic statistical manifold, then $(\widehat{M},\widehat{\nabla },\mathrm{\Lambda },g)$ is called a quaternionic Kähler-like statistical manifold if, for any local basis $\{J_1,J_2,J_3\}$ of Λ, there exist three locally defined $one$-form ${\omega }_1,$ ${\omega }_2,$ ${\omega }_3$ on $\widehat{M}$, such that we have

$$\left({\widehat{\nabla }}_X{J}_{\alpha }\right)Y={\omega }_{\alpha +2}\left(X\right)J_{\alpha +1}Y-{\omega }_{\alpha +1}\left(X\right)J_{\alpha +2}Y$$

for all vector fields on $\widehat{M}$ and for $\alpha \in \{1,2,3\}$.

In 2015, A. D. Vîlcu and G.-E. Vîlcu [201] investigated statistical manifolds with quaternionic settings and suggested a few obvious problems.

The following results were proved by M. S. Lone, M. A. Lone and A. Mihai.

Theorem 84

([202]). Let $({\widehat{M}}^{4m},\widehat{g},\widehat{\nabla },J)$ be a quaternion Kähler-like statistical manifold of dimension $4m$ and M a statistical submanifold of dimension n. Then, we have

$$\begin{array}{cc}\hfill (\tau -K(\pi \left)\right)-& ({\tau }_0-K_0\left(\pi \right))\ge \frac{c}{8}\left(n^2+2n-2\right)-\frac{n^2(n-2)}{4(n-1)}\left[{\parallel H\parallel }^2+{\parallel H^{*}\parallel }^2\right]\hfill \\ & +\frac{c}{4}\sum _{\alpha =1}^3\left\{\parallel P_{\alpha }{\parallel }^2-\frac{1}{2}{\left(\mathrm{Trace}{P}_{\alpha }\right)}^2\right\}+2{\widehat{K}}_{\circ }\left(\pi \right)-2{\widehat{\tau }}_{\circ }\hfill \\ & -\frac{c}{4}\sum _{\alpha =1}^3\left[g(J_{\alpha }{e}_2,e_2)g(J_{\alpha }{e}_1,e_1)+g(e_1,J_{\alpha }{e}_2)g(e_2,J_{\alpha }{e}_1)\right].\hfill \end{array}$$

Moreover, the equalities hold for any $\gamma \in \{n+1,n+2,\cdots ,4m\}$ if, and only if,

$$\begin{array}{cc}& {\sigma }_{11}^{\gamma }+{\sigma }_{22}^{\gamma }={\sigma }_{33}^{\gamma }=\cdots ={\sigma }_{nn}^{\gamma },{\sigma }_{11}^{*\gamma }+{\sigma }_{22}^{*\gamma }={\sigma }_{33}^{*\gamma }=\cdots ={\sigma }_{nn}^{*\gamma },{\sigma }_{ij}^{\gamma }={\sigma }_{ij}^{*\gamma }=0,\hfill \end{array}$$

for $1\le i\ne j\le n$, $\{i,j\}\ne \{1,2\}$.

An important consequence of Theorem 84 is the following.

Theorem 85

( [202]). Let $(\widehat{M},\widehat{g},\widehat{\nabla },J)$ be a quaternion Kähler-like statistical manifold of dimension $4m$ and M be a Lagrangian statistical submanifold of dimension n. If $n\ge 4$ and M satisfies the equality case of the Chen’s first inequality, then it is minimal, i.e., $H=H^{*}=0.$

Remark 45.

In [203], M. Aquib established a first Chen-type inequality for totally real statistical submanifolds in quaternion Kähler-like statistical space forms.

Remark 46.

In [204], M. Kouamou derived the first Chen-type inequality for Lagrangian submanifolds in statistical quaternionic space forms involving $\delta (n_1,\dots ,n_k)$.

Remark 47.

In [205], H. Aytimur, M. Kon, A. Mihai, C. Özgür and K. Takano derived first Chen-type inequalities for holomorphic and totally real submanifolds in Kähler-like statistical manifolds.

29.4. Inequality for Statistical Submanifolds of Sasaki-like Statistical Manifolds

On an almost contact metric manifold $(\overline{M},\varphi ,\xi ,\eta ,g)$, K. Takano [206,207] introduced an additional $(1,1)$-tensor field ${\varphi }^{*}$, which satisfies

$$g({\varphi }^{*}X,Y)=-g(X,\varphi Y)$$

for $X,Y$ tangent to $\overline{M}$, and he named $(\overline{M},\varphi ,\xi ,\eta ,g)$ an almost contact metric manifold of a certain kind. Moreover, K. Takano introduced the notion of a Sasaki-like statistical manifold $(\overline{M},\varphi ,\xi ,\eta ,g,\nabla )$ if it is an almost contact metric, a statistical manifold and it satisfies

$$\begin{array}{cc}\hfill & \left({\nabla }_X\varphi \right)Y=g(X,Y)\xi -\eta \left(Y\right)X,\mathrm{and}\hfill \end{array}$$

(68)

$$\begin{array}{c}\hfill {\nabla }_X\xi =-\varphi X,X,Y\in \mathfrak{X}\left(T\overline{M}\right).\end{array}$$

(69)

In [208], M. E. Aydin, A. Mihai and C. Öz$\stackrel{..}{\mathrm{g}}$ur established first Chen-type inequalities for invariant, anti-invariant statistical submanifolds of a Sasaki-like statistical manifold of a constant curvature. They also discussed the equality cases in [208].

In 2017, H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M. H. Shahid [209] called an almost contact metric structure on $\overline{M}$ a Sasakian structure if it satisfied condition (68). Further, a quadruple $(\nabla ,g,\varphi ,\xi )$ is called a Sasakian statistical structure on $\overline{M}$ if $(\nabla ,g)$ is a statistical structure, $(g,\varphi ,\xi )$ is a Sasakian structure on M and if

$$K_X\varphi Y+\varphi K_{X}Y=0$$

holds for any $X,Y\in \mathfrak{X}\left(T\overline{M}\right)$, where $K_{X}Y={\nabla }_{X}Y-{\nabla }_X^{\circ }Y$.

In [210], M. Aquib, M. N. Boyom, A. H. Alkhaldi and M. H. Shahid derived a statistical version of the first Chen inequality for statistical submanifolds in Sasakian statistical manifolds with a constant curvature. They also discussed the equality case of the inequality and gave some applications of the inequalities they obtained.

Remark 48.

For results in this respect, see also H. Aytimur, A. Mihai and C.Özgür’s article [208].

29.5. Inequality for Space-like Submanifolds in Statistical Manifolds of Para-Kähler Space Forms

A manifold $\overline{M}$ equipped with an almost product structure $\mathcal{P}$ and a pseudo-Riemannian metric $\overline{g}$ is called an almost para-Hermitian manifold if it satisfies

$$g(\mathcal{P}X,\mathcal{P}Y)=-g(X,Y)$$

for all vector fields $X,Y\in \mathfrak{X}\left(\overline{M}\right)$. In particular, if it satisfies $\overline{\nabla }\mathcal{P}=0$ m, then it is called a para-Kähler manifold, where $\overline{\nabla }$ is the Levi–Civita connection of $\overline{M}$.

An almost para-Hermitian-like manifold $(\overline{M},\mathcal{P},\overline{g})$ is a pseudo-Riemannian manifold $(\overline{M},\overline{g})$ equipped with an almost product structure $\mathcal{P}$ satisfying

$$\overline{g}(\mathcal{P}X,Y)+\tilde{g}(X,{\mathcal{P}}^{*}Y)=0$$

for all vector fields $X,Y\in \mathfrak{X}\left(\overline{M}\right)$, where ${\mathcal{P}}^{*}$ is a $(1,1)$-tensor field on $\overline{M}$.

A para-Kähler-like statistical manifold is an almost para-Hermitian-like manifold $(\overline{M},\mathcal{P},\overline{g})$ endowed with a statistical structure $(\overline{\nabla },\overline{g})$, such that $\overline{\mathcal{P}}=0$.

A statistical manifold of a type of para-Kähler space form is a para-Kähler-like statistical manifold where the curvature tensor $\overline{R}$ of the connection $\overline{\nabla }$ satisfies

$$\begin{array}{cc}\hfill \overline{R}(X,Y)Z=& \frac{c}{4}\{\overline{g}(Y,Z)X-\overline{g}(X,Z)Y+\overline{g}(\mathcal{P}Y,Z)\mathcal{P}Z-\overline{g}(\mathcal{P}X,Z)\mathcal{P}Y\hfill \end{array}$$

$$\begin{array}{c}\hfill +\overline{g}(X,\mathcal{P}Y)\mathcal{P}Z-\overline{g}(\mathcal{P}X,Y)\mathcal{P}Z\}\end{array}$$

for any vector fields $X,Y,Z\in \mathfrak{X}\left(\overline{M}\right)$ and a real constant c.

In 2022, S. Decu and S. Haesen derived the first Chen-type inequality for holomorphic and totally real space-like submanifolds in a statistical manifold of a type of para-Kähler space form. For instance, for totally real space-like submanifolds, they proved the following.

Theorem 86

([211]). For an n-dimensional totally real space-like submanifold M of a statistical manifold of a type of para-Kähler space form $(\overline{M},\overline{\nabla },\mathcal{P},\overline{g})$, we have the following inequality:

$$\begin{array}{cc}& \tau -K\left(\pi \right)\ge 2({\tau }_0-K_0\left(\pi \right))+\frac{1}{8}(n-2)(n+1)c-\frac{n^2(n-2)}{4(n-1)}\left({\parallel H\parallel }^2+{\parallel H\parallel }^2\right)\hfill \end{array}$$

for any plane section π of M.

29.6. Inequality for Statistical Submanifolds of Cosymplectic Statistical Space Forms

In [212], Z. Jabeen, M. A. Lone and M. S. Lone defined cosymplectic statistical space forms. They also derived the first Chen-type inequality for a statistical submanifold M in a cosymplectic statistical space form, such that the structure vector field ξ is tangent to M. In the same article, they also determined the necessary and sufficient condition for the equality of the inequality.

29.7. Some Further Results for Statistical Submanifolds and Statistical Submersions

The warped product $B{\times }_{f}F$ of two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ is the product manifold $B\times F$ equipped with the warped product metric $\tilde{g}=g_B+f^2{g}_F$, where f is a smooth function depending only on B. In 2006, L. Todjihounde [213] introduced a suitable dualistic structure on warped product manifolds.

In [214], A. N. Siddiqui, C. Murathan and M. D. Siddiqi studied statistical warped products and they established the corresponding version of the first Chen inequality for submanifolds in a statistical warped product of type $I{\times }_f{\tilde{M}}^m\left(c\right)$, where ${\tilde{M}}^m\left(c\right)$ is an m-dimensional real space form of a constant curvature c.

A $(1,1)$-tensor field ϕ on a Riemannian manifold $(\tilde{M},g)$ satisfying

$$\begin{array}{c}\hfill {\varphi }^2=p\varphi +qI\end{array}$$

(70)

with $p,q\in {\mathbb{N}}^{*}$ is called a metallic structure. A Riemannian manifold $(\tilde{M},g)$ endowed with a metallic structure ϕ is called a metallic Riemannian manifold if the Riemannian metric g is ϕ-compatible, i.e.,

$$\begin{array}{c}\hfill g(\varphi X,Y)=g(X,\varphi Y).\end{array}$$

(71)

Since ϕ is a self-adjoint, after interchanging X by $\varphi X$, we obtain from (71) that

$$g(\varphi X,\varphi Y)=g({\varphi }^{2}X,Y)=pg(X,\varphi Y)+qg(X,Y).$$

Note that if $p=q=1$ in (70), then a metallic structure becomes a golden structure.

A metallic Riemannian manifold $(\tilde{M},g,\varphi )$ is said to be a locally metallic Riemannian manifold if the Levi–Civita connection ∇ of g is a ϕ-connection, that is, $\nabla \varphi =0$.

In [215], O. Bahadir defined and studied metallic-like statistical manifolds. In particular, he derived the first Chen inequality for statistical submanifolds of a metallic-like statistical manifold.

The notion of statistical submersions was introduced in [216] by K. Takano as follows. Let $(M,\nabla ,g)$ and $(B,\tilde{\nabla },\tilde{g})$ be two statistical manifolds. Then, a Riemannian submersion $\varphi :(M,\nabla ,g)\to (B,\tilde{\nabla },\tilde{g})$ is called a statistical submersion if it satisfies

$${\varphi }_{*}{\left({\nabla }_{X}Y\right)}_p={\left({\tilde{\nabla }}_{{\varphi }_{*}X}{\varphi }_{*}Y\right)}_{\varphi \left(p\right)}$$

for basic vector fields $X,Y$ and $p\in M$.

In [217], A. N. Siddiqui, the first author and M. D. Siddiqi derived first Chen-type inequalities for statistical submersions between statistical manifolds.

30. Some Additional Results

In this section, we provide some additional results on first Chen-type inequalities obtained by various authors. In particular, for submanifolds in pseudo-Riemannian manifolds. For general references for submanifolds of pseudo-Riemannian manifolds, we refer to the books in [218,219,220].

30.1. Inequalities for Space-like Hypersurfaces

A generalized Robertson–Walker spacetime $L_1^{n+1}(c,f)$ is a warped product $I{\times }_{f}F$ of a time-like interval I and a real space form F. In [221], N. Poyraz established a first Chen-type inequality for space-like hypersurfaces in a generalized Robertson–Walker spacetime.

30.2. Inequality for Light-like Hypersurfaces

In [222], M. Gülbahar, L. Kiliç and S. Keleş defined screen homothetic light-like hypersurfaces and established the first Chen inequality on a screen homothetic light-like hypersurface of a Lorentzian manifold. Further, N. Poyraz, B. Doğan and E. Yaşar derived in [223] the first Chen inequality for a screen homothetic light-like hypersurface of a Lorentzian space form of a constant sectional curvature endowed with a semisymmetric metric connection.

30.3. Inequalities for Submanifolds in Indefinite Space Forms

P. Zhang, L. Zhang and W.-D. Song [224], and also M. Su and L., Zhang [225], extended Theorem 16 to space-like submanifolds in pseudo-Riemannian space forms using the same idea given in the proof of Theorem 16.

An isometric immersion $\varphi :M_t^n\to {\tilde{M}}_s^m$ from a pseudo-Riemannian manifold of index t to a pseudo-Riemannian manifold of index s is called isotropic if, at the point $p\in M_t^n$, the second fundamental form h satisfies $g\left(h\right(u,u),h(u.u)=\lambda (p)\in \mathbb{R}$ for any unit vector u. An isotropic immersion is one that is isotropic everywhere. In this case, λ is called the isotropy function (see, e.g., [226,227,228,229]).

For isotropic submanifolds in pseudo-Riemannian space forms, M. Mirea [230] derived the following version of the first Chen inequality.

Theorem 87.

Let M be an n-dimensional space-like isotropic submanifold in a pseudo-Riemannian space form ${\tilde{M}}_s^m\left(c\right)$ of a constant curvature c. Then, for any plane section π at a point $p\in M$, we have

$$\tau -K\left(\pi \right)\le \frac{3}{4}{n}^2{\parallel H\parallel }^2-\frac{n^2+2n+4}{4}\lambda +\frac{(n-2)(n+1)}{2}c,$$

where ${\parallel H\parallel }^2=g(H,H)$. Furthermore, the equality holds for a plane section π at a point $p\in M$ if, and only if, there exists an orthonormal basis $\{e_1,e_2\}\subset \pi$, such that $h(e_1,e_2)=0$, where h is the second fundamental form.

In [231], P. Zhang, L. Zhang and W.-D. Song derived the corresponding version of the first Chen inequality for Lagrangian submanifolds in indefinite complex space forms.

30.4. An Optimized First Chen Inequality for Slant Submanifolds in Lorentz–Sasakian Space Forms

A submanifold M tangent to ξ in a Lorentz–Sasakian manifold $(\tilde{M},\varphi ,\xi ,\eta ,g)$ is called a contact slant if for any $p\in M$ and any $X\in T_{p}M$ linearly independent on ${\xi }_p$, the angle between $\varphi \left(X\right)$ and $T_{p}M$ is a constant θ, called the slant angle. A proper contact θ-slant submanifold is said to be a special contact slant if

$$\left({\nabla }_{X}P\right)\left(Y\right)={cos}^2\theta (g(X,Y)\xi +\eta \left(Y\right)X)$$

for vectors $X,Y$ tangent to M (see [232]).

Recall that the first Chen inequality for slant submanifolds in Sasakian space forms was derived by A. Carriazo [142]. In [233], O. Postavaru and I. Mihai derived the first Chen inequality for special contact slant submanifolds in Lorentz–Sasakian space forms.

Theorem 88

([233]). Let M be an $(n+1)$-dimensional special slant submanifold of a $(2n+1)$-dimensional Lorentz–Sasakian space form $\tilde{M}\left(c\right)$ and point $p\in M$ and $\pi \subset T_{p}M$ a plane section orthogonal to ${\xi }_p$. Then, we have

$$\begin{array}{cc}\hfill \tau -infK\left(\pi \right)\le & \frac{n^2(2n-3)}{2(2n+3)}{\parallel H\parallel }^2-\frac{(n+1)(n-2)}{8}(c-3)\hfill \\ & +\frac{3n(c+1)}{8}{cos}^2\theta -\frac{3\mathrm{\Psi }\left(\pi \right)}{4}(c+1)+n{cos}^2\theta ,\hfill \end{array}$$

(72)

where $\mathrm{\Psi }\left(\pi \right)=g{(P{e}_1,e_2)}^2.$

Remark 49.

The necessary and sufficient condition for the equality case of (72) was also determined in [233].

30.5. Inequality for Submanifolds for Warped Product Submanifolds

The first author proved the following optimal inequalities for general warped product submanifolds in real space forms.

Theorem 89

([234]). Let $\varphi :M_1{\times }_f{M}_2\to R^m\left(c\right)$ be an isometric immersion of a warped product into a real space form $R^m\left(c\right)$. Then, we have

$$\frac{\mathsf{\Delta }f}{f}\le \frac{{(n_1+n_2)}^2}{4n_2}{\parallel H\parallel }^2+n_{1}c,n_i=dim{M}_i,i=1,2,$$

where Δ is the Laplacian of $M_1$. The equality holds if, and only if, $\varphi :M_1{\times }_f{M}_2\to R^m\left(c\right)$ is a mixed totally geodesic immersion with $\mathrm{Trace}{h}_1=\mathrm{Trace}{h}_2$, where $\mathrm{Trace}{h}_1$ and $\mathrm{Trace}{h}_2$ denote the trace of h restricted to $M_1$ and $M_2$, respectively.

By combining both methods used in the proofs of Theorem 2 and of Theorem 89, A. Mustafa, C. Özel, A. Pigazzini, R. Kaur and G. Shanker derived in [235] another version of the first Chen inequality for a warped product submanifold $M_1{\times }_f{M}_2$ in a real space form $R^m\left(c\right)$ involving ${\delta }_{M_1}\left(2\right)$ (or ${\delta }_{M_2}\left(2\right)$), ${\parallel H\parallel }^2$ and $(\Delta f)/f$.

30.6. Inequalities for Submanifolds in Locally Product Submanifolds

A locally decomposable Riemannian manifold $\tilde{M}$ is called a ımanifold of an almost constant curvature, denoted by $\tilde{M}(a,b)$, if its curvature tensor $\tilde{R}$ satisfies

$$\begin{array}{cc}\hfill \tilde{R}(X,Y,Z,W)=& a\left\{\right(g(X,W)g(Y,Z)-g(X,Z)g(Y,W))\hfill \\ & +\left(g\right(X,FW\left)g\right(Y,FZ)-g(X,FZ\left)g\right(Y,FW\left)\right)\}\hfill \\ & +b\left\{\right(g(X,FW)g(Y,Z)-g(X,FZ)g(Y,W))\hfill \\ & +\left(g\right(X,W\left)g\right(Y,FZ)-g(X,Z\left)g\right(Y,FW\left)\right)\}\hfill \end{array}$$

for all vector fields $X,Y,Z,W$ tangent to $\tilde{M}$, where F is the almost product structure.

M. Gülbahar, M. M. Tripathi and E. Kiliç [60] studied the invariants $\delta \left(k\right),k=2,\dots ,n-1,$ for n-dimensional submanifolds in a locally decomposable Riemannian manifold. In particular, they derived a version of the first Chen inequality for submanifolds in an almost constant curvature manifold $\tilde{M}(a,b)$.

30.7. Inequalities for Real Hypersurfaces in Some Grassmannians

In [236], M. S. Lone and M. A. Lone derived a version of the first Chen inequality for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians.

30.8. Submanifolds in Sasakian Space Forms with a Tanaka–Webster Connection

For submanifolds tangent to the structure vector field in Sasakian space forms, D. H. Jin and J.W. Lee [237] established a version of the first Chen inequality for submanifolds of a Sasakian space form tangent to the structure vector field in terms of the Tanaka–Webster connection.

30.9. Estimates of the $\delta$-Invariant in Terms of Casorati Curvature and Mean Curvature

F. Casorati [238] introduced, in 1890, what is today called the Casorati curvature (see [239]). In 2019, B. D. Suceavă and M. B. Vajiac [240] provided estimates of Chen’s $\widehat{\delta }$-invariant in terms of Casorati curvature and mean curvature for strictly convex Euclidean hypersurfaces. For results on Casorati curvatures and δ-Casorati curvatures, we refer to the survey paper [241] and the references therein.

30.10. First Chen-Type Inequalities for Affine Hypersurfaces

In [242], C. Scharlach, U. Simon, L. Verstraelen and L. Vrancken defined and investigated an affine version of invariant $\delta \left(2\right)$ for definite centroaffine hypersurfaces in an affine space ${\mathbb{R}}^{n+1}$, $n\ge 3$. In the same article, they established a version of the first Chen inequality for definite centroaffine hypersurfaces. Later, such inequalities have been studied in [147,243,244,245,246], among others.

31. Some Open Problems

In this final section, we would like to pose the following three open problems related to the first Chen inequality.

Open problem 1. One important purpose of the first author to introduce his δ-invariants was to find some applications of Nash’s embedding theorems. In this respect, it would be quite interesting to discover further applications for the various versions of the first Chen inequality.

Open problem 2. The first Chen inequality is an optimal inequality involving the first nontrivial δ-invariant $\delta \left(2\right)$. It is interesting to find applications of the inequalities associated with the other δ-invariants $\delta (n_1,\dots ,n_k)$. In particular, to find applications of inequalities associated with $\delta (2,2)$ and $\delta (2,n-2)$ for n-dimensional submanifolds in various ambient spaces.

Open problem 3. It will be quite interesting to discover new intrinsic and extrinsic invariants that are different from the first author’s δ-invariants and Decu–Haesen–Verstraelen’s δ-Casorati curvature invariants and to establish the corresponding optimal inequalities involving the new invariants for submanifolds in various ambient spaces. Further, it is also important to discover their applications.