Chen-like Inequalities for Submanifolds in Kähler Manifolds Admitting Semi-Symmetric Non-Metric Connections

Abstract

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties.

1. Introduction

The notion of a semi-symmetric linear and metric connection on a differentiable manifold was introduced by A. Friedmann and J.A. Schouten [1] in 1924 and by H.A. Hayden [2] in 1932, respectively.

In 1970, K. Yano dealt with the study of Riemannian manifolds with semi-symmetric metric connections and established their properties (see [3]). N.S. Agashe and M.R. Chafle, in 1992 [4] and 1994 [5], considered semi-symmetric non-metric connections on Riemannian manifolds. They investigated their properties. Moreover induced semi-symmetric, non-metric connections on submanifolds in a Riemannian manifold were studied.

For a non-metric connection, the sectional curvature cannot be defined using the usual definition. Recently, A. Mihai and one of the present authors [6] defined a sectional curvature of a semi-symmetric, non-metric connection.

In 1993 and 1994, in [7,8], B.-Y. Chen defined a new series of curvature invariants, called Chen invariants, and obtained sharp inequalities, well-known as Chen inequalities, for a submanifold in a real space form, which involved intrinsic invariants and the squared mean curvature ${\parallel H\parallel }^2$, the most important extrinsic invariant.

For the Ricci curvature Ric, he established the Chen–Ricci inequality for n-dimensional submanifolds in a Riemannian space form $\overline{M}\left(c\right)$ (see [8]):

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

The equality case was characterized.

As usual, K and $\tau$ can be used to denote the sectional curvature and the scalar curvature, respectively. The Chen invariant ${\delta }_M$ is defined by ${\delta }_M=\tau -infK$.

For an n-dimensional submanifold in a Riemannian space form $\overline{M}\left(c\right)$, this is related to the mean curvature by the Chen first inequality [7]:

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

In the geometry of surfaces in the three-dimensional Euclidean space, a crucial result is the Euler inequality, the inequality between the Gauss curvature and the mean curvature: $K\le {\parallel H\parallel }^2.$ The equality is identical if and only if the surface is totally umbilical.

B.Y. Chen [9] generalized the Euler inequality to submanifolds of arbitrary dimension n in a Riemannian space form $\overline{M}\left(c\right)$, as follows:

$$\rho \le {\parallel H\parallel }^2+c,$$

where $\rho ={\displaystyle \frac{2\tau }{n(n-1)}}$ is the normalized scalar curvature.

Chen’s theory attracted the attention of several authors, who investigated it in various ambient spaces, focusing on special classes of submanifolds.

For instance, A. Oiagă and I. Mihai established in [10] some B.Y. Chen inequalities for slant submanifolds in complex space forms. Afterwards, sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms were obtained in [11]. Also, estimates of the scalar curvature and the k-Ricci curvature were proved in terms of the squared mean curvature.

Next, in [12], J.S. Kim, Y.M. Song, and M.M. Tripathi obtained some B.Y. Chen inequalities for different kinds of submanifolds of generalized complex space forms.

A version of Chen’s inequality for a submanifold of an S-space form, tangent to the structure vector fields of the ambient space, was established, and some applications to the case of slant immersions were obtained from it in [13].

Next, A. Mihai and I. Rădulescu obtained an improved Chen–Ricci inequality in complex space forms in [14,15] a Chen inequality involving the scalar curvature and a Chen–Ricci inequality were obtained for special contact slant submanifolds of Sasakian space forms.

In 2022, Y. Li et al. [16] improved the Chen inequalities for submanifolds in generalized Sasakian space forms. Also, in [17], M. Mohammed established the Chen first inequality and Chen–Ricci inequality, respectively, for bi-slant submanifolds in Kenmotsu space forms.

Recently, various curvature inequalities involving Ricci and the scalar curvature of the horizontal and vertical distributions of a quasi bi-slant Riemannian submersion from complex space forms onto a Riemannian manifold were obtained in [18].

There are many papers studying Chen inequalities on submanifolds in various ambient spaces admitting a semi-symmetric metric connection; see [19,20,21,22].

Because the sectional curvature of a semi-symmetric, non-metric connection cannot be defined using the standard definition, in [6] the authors defined a sectional curvature on a Riemannian manifold admitting such a connection. Thus, our paper is the first to study Chen inequalities for submanifolds in complex space forms with a semi-symmetric, non-metric connection.

In this paper, we deal with submanifolds of complex space forms admitting a semi-symmetric, non-metric connection, in terms of the sectional curvature defined in [6]. The present paper is structured as follows: in Section 2, the basic properties of a semi-symmetric non-metric connection are recalled. In the following section, we prove a Chen-Ricci inequality for submanifolds in Kähler manifolds admitting a semi-symmetric non-metric connection. Some applications are derived. In Section 4, a Chen first inequality for such submanifolds is established. Next, in Section 5, we consider certain submanifolds in complex space forms admitting a semi-symmetric, non-metric connection for which a generalized Euler inequality is proven.

2. Preliminaries

In this section, we recall the notations and fundamental results and formulas for later use.

We consider a Riemannian manifold $(\overline{M},g)$ of dimension m and a linear connection $\overline{\nabla }$ on $\overline{M}$. Its torsion tensor field $\overline{T}$ is defined by

$$\overline{T}(X,Y)={\overline{\nabla }}_{X}Y-{\overline{\nabla }}_{Y}X-[X,Y],$$

(1)

for any vector fields $X,Y$ tangent to $\overline{M}$.

The connection $\overline{\nabla }$ is called a semi-symmetric connection if $\overline{T}$ is given by

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

(2)

with $\omega$ being an 1-form associated with the vector field P on $\overline{M}$, i.e., satisfying $\omega \left(X\right)=g(X,P)$.

The semi-symmetric connection $\overline{\nabla }$ is called a metric connection if the Riemannian metric g is parallel, i.e., $\overline{\nabla }g=0$. Otherwise, i.e., $\overline{\nabla }g\ne 0$, $\overline{\nabla }$ is said to be a semi-symmetric, non-metric connection.

It is known (see [4]) that a semi-symmetric non-metric connection $\overline{\nabla }$ on $\overline{M}$ is expressed by

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X}Y={\stackrel{\circ }{\tilde{\nabla }}}_{X}Y+\omega \left(Y\right)X,\end{array}$$

for all vector fields $X,Y$ tangent to $\overline{M}$, where $\stackrel{\circ }{\tilde{\nabla }}$ is the Levi–Civita connection corresponding to g.

Next, we will consider $(\overline{M},g)$ a Riemannian manifold admitting a semi-symmetric, non-metric connection $\overline{\nabla }$ and the Levi–Civita connection $\stackrel{\circ }{\tilde{\nabla }}$. Let M be an n-dimensional submanifold of $\overline{M}$.

If $\overline{R}$ and $\stackrel{\circ }{\tilde{R}}$ are the curvature tensors of the Riemannian manifold $\overline{M}$ with respect to $\overline{\nabla }$ and $\stackrel{\circ }{\tilde{\nabla }}$, respectively, using the notation $R(X,Y,Z,W)=g\left(R\right(X,Y)Z,W)$, $\overline{R}$ can be written as [4]

$$\begin{array}{c}\hfill \overline{R}(X,Y,Z,W)=\stackrel{\circ }{\tilde{R}}(X,Y,Z,W)+s(X,Z)g(Y,W)-s(Y,Z)g(X,W),\end{array}$$

(3)

for any vector fields $X,Y,Z,W$ tangent to $\overline{M}$, where s is a $(0,2)$-tensor expressed by

$$s(X,Y)=\left({\stackrel{\circ }{\tilde{\nabla }}}_X\omega \right)Y-\omega \left(X\right)\omega \left(Y\right).$$

We denote, using $\lambda$, the trace of s.

The Gauss formulae for $\overline{\nabla }$ and $\stackrel{\circ }{\tilde{\nabla }}$, respectively, are expressed by

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),{\stackrel{\circ }{\tilde{\nabla }}}_{X}Y={\stackrel{\circ }{\nabla }}_{X}Y+\stackrel{\circ }{h}(X,Y),\end{array}$$

for all vector fields $X,Y$ on the submanifold M of dimension n; $\stackrel{\circ }{h}$ is the second fundamental form of M and h is a $(0,2)$-tensor on M. It was shown in [5] that $\stackrel{\circ }{h}=h$.

Recall that an even dimensional Riemannian manifold $(\overline{M},g)$ is called an almost-Hermitian manifold if there exists an almost complex structure J$(\overline{M},g)$ (i.e., an anti-involutive $(1,1)$-tensor field satisfying $J^{2}X=-X$) such that

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

for all the vector fields $X,Y$ on $\overline{M}$.

An almost-Hermitian manifold is a Kähler manifold if $\stackrel{\circ }{\tilde{\nabla }}J=0$, i.e., the almost complex structure J is parallel with respect to the Levi–Civita connection $\stackrel{\circ }{\tilde{\nabla }}$.

A two-plane section $\pi$ is called holomorphic if $J\left(\pi \right)=\pi$, i.e., it is invariant by J. The sectional curvature of a holomorphic two-plane section is called the holomorphic sectional curvature.

A Kähler manifold $(\overline{M},J,g)$ with a constant holomorphic sectional curvature c is called a complex space form; it is denoted by $\overline{M}\left(c\right)$. In this case, its curvature tensor field is expressed by

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

(4)

for all vector fields $X,Y,Z$.

If $\overline{M}\left(c\right)$ is a $2m$-dimensional complex space form endowed with a semi-symmetric non-metric connection, then, using (3) and (4), the curvature tensor field $\overline{R}$ takes the form

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

(5)

Let M be an n-dimensional submanifold. We decompose the vector field P on M into its tangent and normal components $P^{\top }$ and $P^{\perp }$; then, we write $P=P^{\top }+P^{\perp }$.

The Gauss equation for the semi-symmetric non-metric connection is (see [5])

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)& =R(X,Y,Z,W)+g\left(h\right(X,Z),h(Y,W\left)\right)-g\left(h\right(X,W),h(Y,Z\left)\right)\hfill \\ \hfill & +g(P^{\perp },h(Y,Z))g(X,W)-g(P^{\perp },h(X,Z))g(Y,W),\hfill \end{array}$$

(6)

for any $X,Y,Z,W\in \mathsf{\Gamma }\left(TM\right)$.

In [6], A. Mihai and the first author defined a sectional curvature of a non-metric connection as follows.

Let $\pi \subset T_{p}M$ be a two-plane section at $p\in M$ and $\{e_1,e_2\}$ an orthonormal basis of $\pi$. Because $R(X,Y,Z,W)\ne R(X,Y,W,Z)$, the sectional curvature $K\left(\pi \right)$ cannot be defined using the standard definition. The sectional curvature $K\left(\pi \right)$ of $\pi$ with respect to the induced connection ∇ is expressed by (see [6])

$$\begin{array}{c}\hfill K\left(\pi \right)={\displaystyle \frac{1}{2}}[R(e_1,e_2,e_2,e_1)+R(e_2,e_1,e_1,e_2)].\end{array}$$

(7)

Obviously, the above definition does not depend on the orthonormal basis.

If $\{e_1,e_2,\cdots ,e_n\}$ is an orthonormal basis of the tangent space $T_{p}M$ at $p\in M$, then the scalar curvature $\tau$ at p is given by

$$\begin{array}{c}\hfill \tau \left(p\right)=\sum _{1\le i<j\le n}{K}_{ij},\end{array}$$

(8)

where $K_{ij}$ is the sectional curvature of the two-plane section spanned by vectors $e_i$ and $e_j$.

The Chen first invariant is expressed by

$${\delta }_M\left(p\right)=\tau \left(p\right)-infK\left(p\right),$$

(9)

with $(infK)\left(p\right)=inf\{K\left(\pi \right);\pi \subset T_{p}M,dim\pi =2\}$.

For a k-plane section L in $T_{p}M$ and $X\in L$, a unit vector, we choose an orthonormal basis $\{e_1,\dots ,e_k\}$ of L with $e_1=X$. Then, the Ricci curvature $Ri{c}_L$ of L at X, called the k-Ricci curvature, is defined by

$$Ri{c}_L\left(X\right)=K_{12}+K_{13}+\cdots +K_{1k}.$$

(10)

Recall that the mean curvature vector $H\left(p\right)$ at $p\in M$ is expressed by

$$H\left(p\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}h(e_i,e_i).$$

(11)

Using $h_{ij}^r=g(h(e_i,e_j),e_r)$, $i,j=1,\dots ,n,r\in \{n+1,\dots ,2m\}$, we denote the components of the second fundamental form h. Then, the squared norm of the second fundamental form h is

$${\parallel h\parallel }^2=\sum _{r=n+1}^{2m}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2.$$

Using the definition of the 1-form $\omega$, we have

$$\omega \left(H\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n\omega \left(h(e_i,e_i)\right)=g(P^{\perp },H).$$

(12)

For any $X\in \mathsf{\Gamma }\left(TM\right)$, we decompose $JX$ into its tangential component $PX$ and normal component $FX$, respectively, i.e., $JX=PX+FX$. We denote

$${\parallel P\parallel }^2=\sum _{i,j=1}^n{g}^2(P{e}_i,e_j).$$

We present an algebraic lemma that will be used in Section 3.

Lemma 1 

([23]). The function $f(x_1,x_2,\cdots ,x_n)$, $n\ge 3$, on ${\mathbb{R}}^n$ is given by

$$\begin{array}{c}\hfill f(x_1,x_2,\cdots ,x_n)=(x_1+x_2)\sum _{i=3}^n{x}_i+\sum _{3\le i<j\le n}{x}_i{x}_j.\end{array}$$

If $x_1+x_2+\cdots +x_n=(n-1)a$, then one has

$$f(x_1,x_2,\cdots ,x_n)\le {\displaystyle \frac{(n-1)(n-2)}{2}}{a}^2.$$

The equality holds if and only if $x_1+x_2=x_3=\cdots =x_n=a$.

In the following sections, we will consider submanifolds in a complex space form endowed with a semi-symmetric non-metric connection.

This is the first study of Chen inequalities for such submanifolds.

3. Chen–Ricci Inequality for Arbitrary Submanifolds in Complex Space Forms Admitting a Semi-Symmetric, Non-Metric Connection

In [8], B.-Y. Chen established an inequality relating the Ricci curvature and the squared mean curvature for a Riemannian submanifold of dimension n in a Riemannian space form with constant sectional curvature c,

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

The above inequality is named the Chen–Ricci inequality.

Furthermore, in an article by K. Matsumoto et al. [24], a corresponding Chen–Ricci inequality for arbitrary submanifolds in complex space forms was proven.

In the present section, we prove a Chen–Ricci inequality for submanifolds in complex space forms admitting a semi-symmetric, non-metric connection. Certain corollaries are derived.

Theorem 1. 

Let M be an n-dimensional submanifold of a $2m$-dimensional complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection. Then, the following hold:

(1) 

The Ricci curvature of any unit vector $X\in T_{p}M$ satisfies:

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{n^2}{4}}{\parallel H\parallel }^2+(n-1){\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]-{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))].\hfill \end{array}$$

(13)

(2) 

If $H\left(p\right)=0$, then a unit tangent vector X at p satisfies the equality case of (13) if and only if $X\in N_p$, where $N_p=\{X\in T_{p}M|h(X,Y)=0,\forall Y\in T_{p}M\}$ is the kernel of the second fundamental form.

(3) 

The equality case of (13) holds identically for all unit tangent vectors at p if and only if one of the following holds:

(i) 

p is a totally geodesic point;

(ii) 

$n=2$ and p is a totally umbilical point.

Proof. 

(1) Let p be a point in M, X a unit vector tangent to M at p and $\{e_1,\cdots ,e_n,$$e_{n+1},\cdots ,e_{2m}\}$ an orthonormal basis of $T_p\overline{M}\left(c\right)$, such that $e_1,\cdots ,e_n$ are tangent to M and $e_1=X$. Then, use the following definition:

$$\begin{array}{cc}\hfill Ric\left(X\right)=& \sum _{j=2}^{n}K(e_1\wedge e_j).\hfill \end{array}$$

(14)

The Gauss equation implies

$$\begin{array}{cc}\hfill R(e_1,e_j,e_j,e_1)& ={\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{g}^2(J{e}_1,e_j)-s(e_j,e_j)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2]-g(P^{\perp },h(e_j,e_j)).\hfill \end{array}$$

(15)

Similarly, interchanging $e_1$ and $e_j$, we can obtain

$$\begin{array}{cc}\hfill R(e_j,e_1,e_1,e_j)& ={\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{g}^2(J{e}_j,e_1)-s(e_1,e_1)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2]-g(P^{\perp },h(e_1,e_1)).\hfill \end{array}$$

(16)

Recall that

$$\begin{array}{cc}\hfill K(e_1\wedge e_j)& ={\displaystyle \frac{1}{2}}[R(e_1,e_j,e_j,e_1)+R(e_j,e_1,e_1,e_j)].\hfill \end{array}$$

(17)

Therefore, using (15)–(17), we can obtain

$$\begin{array}{cc}\hfill K(e_1\wedge e_j)& ={\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{g}^2(J{e}_1,e_j)-{\displaystyle \frac{1}{2}}[s(e_j,e_j)+s(e_1,e_1)]\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[g(P^{\perp },h(e_j,e_j))+g(P^{\perp },h(e_1,e_1))].\hfill \end{array}$$

(18)

If we substitute the Equation (18) in (14), we can obtain

$$\begin{array}{cc}\hfill Ric\left(X\right)& =(n-1){\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2-{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))]+\sum _{r=n+1}^{2m}\sum _{j=2}^n[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2].\hfill \end{array}$$

(19)

From the above equation, we can obtain

$$\begin{array}{cc}\hfill & Ric\left(X\right)-(n-1){\displaystyle \frac{c}{4}}-{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2+{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))]\hfill \\ \hfill & \le \sum _{r=n+1}^{2m}\sum _{j=2}^n{h}_{11}^r{h}_{jj}^r.\hfill \end{array}$$

(20)

Obviously, one has

$$ab\le {\displaystyle \frac{1}{4}}{(a+b)}^2;$$

the equality holds if and only if $a=b$.

Setting $a=h_{11}^r$ and $b=h_{22}^r+\dots +h_{nn}^r$, we can obtain

$$h_{11}^r(h_{22}^r+\dots +h_{nn}^r)\le {\displaystyle \frac{1}{4}}{(h_{11}^r+h_{22}^r+\dots +h_{nn}^r)}^2.$$

(21)

It follows that

$$\sum _{r=n+1}^{2m}\sum _{j=2}^n{h}_{11}^r{h}_{jj}^r\le {\displaystyle \frac{n^2}{4}}{\parallel H\parallel }^2.$$

Then, (20) implies the inequality that is to be proved.

(2) Let $X\in T_{p}M$ be a unit vector satisfying the equality case of (13). Then, from (20) and (21), we can obtain

$$\left\{\begin{array}{c}{h}_{1i}^r=0,2\le i\le n,\forall r\in \{n+1,\dots ,2m\},\hfill \\ h_{11}^r=h_{22}^r+\cdots +h_{nn}^r,\forall r\in \{n+1,\dots ,2m\}.\hfill \end{array}\right.$$

Thus, since $H\left(p\right)=0$, it follows that $h_{1j}^r=0$, for any $j\in \{1,\cdots ,n\}$, $r\in \{n+1,\cdots ,2m\}$, i.e., $X\in N_p$.

(3) The equality of inequality (13) holds for any unit vectors in $T_{p}M$ if and only if

$$\left\{\begin{array}{c}{h}_{ij}^r=0,1\le i\ne j\le n,r\in \{n+1,\cdots ,2m\},\hfill \\ h_{11}^r+\cdots +h_{nn}^r-2h_{ii}^r=0,i\in \{1,\cdots ,n\},r\in \{n+1,\cdots ,2m\}.\hfill \end{array}\right.$$

We distinguish two cases:

• (i) $n\ne 2$; then $h_{ij}^r=0$, $\forall i,j\in \{1,\dots ,n\}$, $\forall r\in \{n+1,\dots ,2m\}$.
  It follows that p is a totally geodesic point.
• (ii) $n=2$; then p is a totally umbilical point. □

Next, we will state the corresponding Chen–Ricci inequalities for some classes of submanifolds in the Hermitian manifolds.

We recall these standard definitions.

Let $\overline{M}$ be a Hermitian manifold, and M a submanifold of $\overline{M}$, and using J, denote the canonical, almost complex structure on $\overline{M}$.

Then, M is said to be a complex submanifold if $J\left(T_{p}M\right)=T_{p}M$ for all $p\in M$.

If $J\left(T_{p}M\right)\subset T_p^{\perp }M$ for all $p\in M$, where $T_{p}M$ (respectively $T_p^{\perp }M$) is the tangent (respectively, the normal) vector space of M in p, then M is said to be a totally real submanifold. A totally real submanifold is called Lagrangian if it has the maximum dimension $n=m$.

For any non-null vector $X\in T_{p}M$, the angle $\theta \left(X\right)$ between $JX$ and the tangent space $T_{p}M$ is called the Wirtinger angle of X.

The submanifold M is said to be a slant submanifold if the Wirtinger angle $\theta \left(X\right)$ is constant (independent of the choice of $p\in M$ and $X\in T_{p}M$).

Obviously, the complex submanifolds and the totally real submanifolds are slant submanifolds ($\theta =0$ and $\theta ={\displaystyle \frac{\pi }{2}}$, respectively).

Corollary 1. 

Let M be an n-dimensional ($n\ge 2$) complex submanifold of a $2m$-dimensional complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection. Then

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le (n-1){\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}(n-2)g(P^{\perp },h(X,X)),\hfill \end{array}$$

(22)

for each unit vector $X\in T_{p}M$.

Proof. 

It is known that the complex submanifold of a Kähler manifold is minimal. □

Corollary 2. 

Let M be an n-dimensional ($n\ge 2$) totally real submanifold of a $2m$-dimensional complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection. Then

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{n^2}{4}}{\parallel H\parallel }^2+(n-1){\displaystyle \frac{c}{4}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))],\hfill \end{array}$$

(23)

for each unit vector $X\in T_{p}M$.

Proof. 

On a totally real submanifold, $PX=0$. □

Corollary 3. 

Let M be an n-dimensional ($n\ge 2$) slant submanifold with the Wirtinger angle θ of a $2m$-dimensional complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection. Then:

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{n^2}{4}}{\parallel H\parallel }^2+(n-1){\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4}}{cos}^2\theta \hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))],\hfill \end{array}$$

(24)

for each unit vector $X\in T_{p}M$.

Proof. 

On a slant submanifold $\parallel PX\parallel =cos\theta$. □

We recall the definition of a H-umbilical submanifold.

A Lagrangian submanifold of a Kähler manifold $\overline{M}$ is a H-umbilical submanifold if the second fundamental form h is given by:

$$\begin{array}{c}\hfill h(e_1,e_1)=\alpha J{e}_1,h(e_2,e_2)=\dots =h(e_n,e_n)=\beta J{e}_1\\ \hfill h(e_1,e_j)=\beta e_j,h(e_j,e_k)=0,j\ne k,j,k=2,\dots n,\end{array}$$

(25)

for some smooth functions $\alpha$ and $\beta$ with respect to some suitable orthornormal local frame field.

Considering the optimization technique used by Deng [25], one can obtain the following improved Chen–Ricci inequality for Lagrangian submanifolds in complex space forms.

Theorem 2. 

Let M be an n-dimensional ($n\ge 2$) Lagrangian submanifold of a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection.

Then

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{n-1}{4}}{[n\parallel H\parallel }^2+c]+\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))],\hfill \end{array}$$

(26)

for any unit vector $X\in T_{p}M$.

The equality holds for any unit tangent vector at p if and only if one of the following holds:

(i) 

p is a totally geodesic point;

(ii) 

$n=2$ and p is a H-umbilical point with $\alpha =3\beta$.

Proof. 

Let $p\in M$, $X\in T_{p}M$ unit and $\{e_1=X,\dots ,e_n\}$ an orthonormal basis of $T_{p}M$. Then $\{e_{n+1}=e_{1^{*}}=J{e}_1,\dots ,e_{2n}=e_{n^{*}}=J{e}_n\}$ is an orthonormal basis of $T_p^{\perp }M$. We denote by $h_{ij}^r=g(h(e_i,e_j),J{e}_r)$, $i,j,r=1,\dots ,n$.

The Equation (19) can be rewritten in the form

$$\begin{array}{cc}\hfill & Ric\left(X\right)-(n-1){\displaystyle \frac{c}{4}}-{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2+{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))]\hfill \\ \hfill & =\sum _{r=1}^n\sum _{j=2}^n[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2].\hfill \end{array}$$

(27)

It is known that, on a Lagrangian submanifold, $h_{ij}^r=h_{ir}^j=h_{jr}^i$, $\forall i,j,r=1,\dots ,n$.

Then, the above equation implies the following:

$$\begin{array}{cc}\hfill & Ric\left(X\right)-(n-1){\displaystyle \frac{c}{4}}-{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2+{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))]\hfill \\ \hfill & \le \sum _{r=1}^n\sum _{j=2}^n{h}_{11}^r{h}_{jj}^r-\sum _{j=2}^n{h}_{11}^j-\sum _{j=2}^n{h}_{jj}^1.\hfill \end{array}$$

(28)

By using the same arguments as in [25], we can obtain

$$\begin{array}{cc}\hfill & Ric\left(X\right)-(n-1){\displaystyle \frac{c}{4}}-{\displaystyle \frac{3c}{4}}{\parallel PX\parallel }^2+{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))]\hfill \\ \hfill & \le {\displaystyle \frac{n(n-1)}{4}}{\parallel H\parallel }^2,\hfill \end{array}$$

(29)

which is equivalent to the desired inequality. □

Example 1. 

The Whitney 2-sphere in ${\mathbb{C}}^2$ is the image of the Lagrangian immersion of the unit sphere $S^2$, centered at the origin, in ${\mathbb{R}}^3$ to ${\mathbb{C}}^2$, given by

$$\varphi :S^2\to {\mathbb{C}}^2,$$

$$\begin{array}{c}\hfill \varphi (x_1,x_2,x_3)={\displaystyle \frac{1}{1+x_3^2}}(x_1,x_1{x}_3,x_2,x_2{x}_3).\end{array}$$

(30)

This satisfies the equality case of (26).

From Theorem 2, we derive the following result.

Corollary 4. 

Let M be an n-dimensional ($n\ge 2$) Lagrangian submanifold of a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection. If

$$\begin{array}{cc}\hfill Ric\left(X\right)=& {\displaystyle \frac{n-1}{4}}{[n\parallel H\parallel }^2+c]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\lambda +(n-2)s(X,X)]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[n\omega \left(H\right)+(n-2)g(P^{\perp },h(X,X))],\hfill \end{array}$$

(31)

for any vector field X tangent to M, then either M is a totally geodesic submanifold in $\overline{M}\left(c\right)$ or $n=2$ and M is a H-umbilical submanifold of $\overline{M}\left(c\right)$ with $\alpha =3\beta$.

4. Chen First Inequality for Submanifolds in Complex Space Forms Endowed with a Semi-Symmetric, Non-Metric Connection

In this section, we establish the Chen first inequality for submanifolds in complex space forms admitting a semi-symmetric non-metric connection.

Theorem 3. 

Let M be an n-dimensional $(n\ge 3)$ submanifold isometrically immersed in a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection, $p\in M$, $\pi \subset T_{p}M$ a 2-plane section and denote by ${\psi }^2\left(\pi \right)=g^2(J{e}_1,e_2)$. Then,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le (n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}{\{\parallel P\parallel }^2-2{\psi }^2\left(\pi \right)\}\hfill \\ \hfill & -(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)+{\displaystyle \frac{1}{2}}\mathrm{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathrm{trace}\left(h_{|\pi }\right),P)\hfill \\ \hfill & +{\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2.\hfill \end{array}$$

(32)

Moreover, the equality case of (32) holds if and only if the shape operators have the following forms:

$$A_{e_r}=\left(\begin{array}{ccccc}{h}_{11}^r& h_{12}^r& 0& \cdots & 0\\ h_{12}^r& h_{22}^r& 0& \cdots & 0\\ 0& 0& h_{33}^r& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & h_{nn}^r\end{array}\right),$$

with $h_{11}^r+h_{22}^r=h_{33}^r=\cdots =h_{nn}^r$, for all $r\in \{n+1,\dots ,2m\}$.

Proof. 

Consider $\overline{M}\left(c\right)$ to be a $2m$-dimensional complex space form admitting a semi-symmetric non-metric connection $\overline{\nabla }$ and M to be a submanifold of $\overline{M}\left(c\right)$ with $dimM=n$.

Let $p\in M$, $\pi \subset T_{p}M$ be a two-plane section. We take $\{e_1,\dots ,e_n\}$ as the orthonormal basis of the tangent space $T_{p}M$ and $\{e_{n+1},\dots ,e_{2m}\}$ as the orthonormal basis of $T_p^{\perp }M$, with $\pi =span\{e_1,e_2\}$.

We recall the following formula:

$$\begin{array}{c}\hfill 2\tau =\sum _{1\le i\ne j\le n}K(e_i\wedge e_j).\end{array}$$

(33)

If we use $X=W=e_i$ and $Y=Z=e_j$, $i,j=1,\cdots ,n$, the Gauss equation implies the following:

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =\sum _{1\le i\ne j\le n}\overline{R}(e_i,e_j,e_j,e_i)+2\sum _{r=n+1}^{2m}\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & -(n-1)\sum _{j=1}^{n}g(P^{\perp },h(e_j,e_j)).\hfill \end{array}$$

(34)

We calculate $\overline{R}(e_i,e_j,e_j,e_i)$, taking into account the expression (5) of the curvature tensor, where we $X=W=e_i$, $Y=Z=e_j$ are used for $i,j=1,\cdots n,i\ne j$.

$$\begin{array}{cc}\hfill \overline{R}(e_i,e_j,e_j,e_i)& ={\displaystyle \frac{c}{4}}\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\hfill \\ \hfill & +g(e_i,J{e}_j)g(e_j,e_i)-g(e_j,J{e}_j)g(J{e}_i,e_i)\hfill \\ \hfill & +2g(e_i,J{e}_j)g(J{e}_j,e_i)\}-s(e_j,e_j)g(e_i,e_i).\hfill \end{array}$$

(35)

Then,

$$\begin{array}{c}\hfill \overline{R}(e_i,e_j,e_j,e_i)={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_i,e_j)]-s(e_j,e_j).\end{array}$$

(36)

If we substitute (36) in the Equation (34), we can obtain the following:

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =[n(n-1)+3\sum _{1\le i\ne j\le n}{g}^2(J{e}_i,e_j)]{\displaystyle \frac{c}{4}}-(n-1)\lambda \hfill \\ \hfill & +2\sum _{r=n+1}^{2m}\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & -n(n-1)\omega \left(H\right),\hfill \end{array}$$

(37)

where $\lambda =$ trace s.

Let $\pi =$ span$\{e_1,e_2\}$. If we use $X=W=e_1$, $Y=Z=e_2$, we can obtain the following from the Gauss equation:

$$\begin{array}{cc}\hfill R(e_1,e_2,e_2,e_1)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_2)-s(e_2,e_2)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-g(P^{\perp },h(e_2,e_2)).\hfill \end{array}$$

(38)

Interchanging $e_1$ and $e_2$, we find

$$\begin{array}{cc}\hfill R(e_2,e_1,e_1,e_2)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_2)]-s(e_1,e_1)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-g(P^{\perp },h(e_1,e_1)).\hfill \end{array}$$

(39)

So, from (7), (38) and (39), we have

$$\begin{array}{cc}\hfill K\left(\pi \right)& =[1+3g^2(J{e}_1,e_2)]{\displaystyle \frac{c}{4}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\mathrm{trace}\left(s_{|\pi }\right)-{\displaystyle \frac{1}{2}}g(\mathrm{trace}\left(h_{|\pi }\right),P)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

(40)

Since ${\psi }^2\left(\pi \right)=g^2(J{e}_1,e_2)$, we obtain

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =(n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}{\{\parallel P\parallel }^2-2{\psi }^2\left(\pi \right)\}\hfill \\ \hfill & -(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}trace\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(trace\left(h_{|\pi }\right),P)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}\{(h_{11}^r+h_{22}^r)\sum _{3\le i\le n}{h}_{ii}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{3\le j\le n}{\left(h_{1j}^r\right)}^2-\sum _{3\le j\le n}{\left(h_{2j}^r\right)}^2-\sum _{3\le i<j\le n}{\left(h_{ij}^r\right)}^2\}.\hfill \end{array}$$

(41)

Then,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le (n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}{\{\parallel P\parallel }^2-2{\psi }^2\left(\pi \right)\}\hfill \\ \hfill & -(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}trace\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(trace\left(h_{|\pi }\right),P)\hfill \\ \hfill & +\sum _{r=n+1}^{2m}\left\{(h_{11}^r+h_{22}^r)\sum _{3\le i\le n}{h}_{ii}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\right\}.\hfill \end{array}$$

(42)

From Lemma 1, we have

$$(h_{11}^r+h_{22}^r)\sum _{3\le i\le n}{h}_{ii}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\le {\displaystyle \frac{n-2}{2(n-1)}}{(h_{11}^r+\dots +h_{nn}^r)}^2.$$

(43)

The equality case is equivalent to

$$h_{11}^r+h_{22}^r=h_{33}^r=\cdots =h_{nn}^r.$$

Next, taking into account (42) and (43), we can obtain

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =(n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}{\{\parallel P\parallel }^2-2{\psi }^2\left(\pi \right)\}\hfill \\ \hfill & -(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}trace\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(trace\left(h_{|\pi }\right),P)\hfill \\ \hfill & +{\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2.\hfill \end{array}$$

(44)

We see that the equality holds if and only if

$h_{1j}^r=h_{2j}^r=0,\forall j=1,n$, $r\in \{n+1,\dots ,2m\}$,

$h_{ij}^r=0,\forall 1\le i\ne j\le n$, $r\in \{n+1,\dots ,2m\},$

$h_{11}^r+h_{22}^r=h_{33}^r=\cdots =h_{nn}^r,$$r\in \{n+1,\dots ,2m\}$.

Then, the shape operators have the desired forms. □

Theorem 3 implies the following corollaries. In their proofs, we use the same arguments as in the proofs of Corolaries 1–3.

Corollary 5. 

Let M be an n-dimensional $(n\ge 3)$ complex submanifold that is isometrically immersed in a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection, and $p\in M$ and $\pi \subset T_{p}M$ be a 2-plane section. Then,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le (n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}[n-2{\psi }^2\left(\pi \right)]-(n-1){\displaystyle \frac{\lambda }{2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathrm{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathrm{trace}\left(h_{|\pi }\right),P).\hfill \end{array}$$

(45)

Corollary 6. 

Let M be an n-dimensional $(n\ge 3)$, totally real submanifold that is isometrically immersed in a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection, and $p\in M$ and $\pi \subset T_{p}M$ be a 2-plane section. Then,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le (n-2)(n+1){\displaystyle \frac{c}{8}}-(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathrm{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathrm{trace}\left(h_{|\pi }\right),P)\hfill \\ \hfill & +{\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2.\hfill \end{array}$$

(46)

Proof. 

On a totally real submanifold $\psi \left(\pi \right)=0$. □

Corollary 7. 

Let M be an n-dimensional $(n\ge 3)$ slant submanifold that is isometrically immersed in a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric, non-metric connection, and $p\in M$ and $\pi \subset T_{p}M$ be a 2-plane section. Then,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le (n-2)(n+1){\displaystyle \frac{c}{8}}+{\displaystyle \frac{3c}{8}}[n{cos}^2\theta -2{\psi }^2\left(\pi \right)]\hfill \\ \hfill & -(n-1){\displaystyle \frac{\lambda }{2}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathrm{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathrm{trace}\left(h_{|\pi }\right),P)\hfill \\ \hfill & +{\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2.\hfill \end{array}$$

(47)

5. A Generalized Euler Inequality

It is well-known that on a surface M in the three-dimensional Euclidean space, we have the following inequality between the Gauss curvature K and the mean curvature $\parallel H\parallel$:

$$K\le {\parallel H\parallel }^2,$$

known as the Euler inequality, which is equal if and only if M is a totally umbilical surface. Using a theorem of Meusnier, M is either an open portion of a Euclidean plane or an open portion of a Euclidean sphere.

B.Y. Chen [9] generalized the above inequality for arbitrary n-dimensional submanifolds M in real space forms with constant sectional curvature c:

$${\parallel H\parallel }^2\ge {\displaystyle \frac{2\tau }{n(n-1)}}-c,$$

where $dimM=n$.

The equality holds at any $p\in M$ if and only if M is totally umbilical.

Next, we establish a generalized Euler inequality for Kählerian slant submanifolds in a complex space form admitting a semi-symmetric non-metric connection.

A proper slant submanifold M ($\theta \ne 0,{\displaystyle \frac{\pi }{2}})$ of a complex space form $\overline{M}\left(c\right)$ is called a Kählerian slant submanifold [26] if the endomorphism P is parallel with respect to the Levi–Civita connection, i.e., $\stackrel{\circ }{\nabla }P=0$.

It is known that the above condition is equivalent to

$$A_{FX}Y=A_{FY}X,\forall X,Y\in \mathsf{\Gamma }\left(TM\right).$$

We will obtain a generalized Euler inequality for Kählerian slant submanifolds.

Theorem 4. 

Let $\overline{M}\left(c\right)$ be a $2n$-dimensional complex space form admitting a semi-symmetric non-metric connection $\overline{\nabla }$ and M an n-dimensional $(n\ge 2)$ Kählerian slant submanifold of $\overline{M}\left(c\right)$. Then,

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge {\displaystyle \frac{2(n+2)}{n^2(n-1)}}\tau -{\displaystyle \frac{n+2}{n}}{\displaystyle \frac{c}{4}}\hfill \\ \hfill & +3{\displaystyle \frac{n+2}{n(n-1)}}{\displaystyle \frac{c}{4}}{cos}^2\theta \hfill \\ \hfill & +{\displaystyle \frac{n+2}{n^2}}\lambda +{\displaystyle \frac{n+2}{n}}\omega \left(H\right).\hfill \end{array}$$

(48)

In addition, the equality holds identically if and only if M is a H -umbilical submanifold.

Proof. 

Consider a $2n$-dimensional complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection $\overline{\nabla }$ and M to be an n-dimensional Kählerian slant submanifold of $\overline{M}\left(c\right)$.

Let $p\in M$ and $\pi \subset T_{p}M$ be a 2-plane section, $\{e_1,\dots ,e_n\}$ serve as an orthonormal basis of the tangent space $T_{p}M$, and $\{e_{n+1},\dots ,e_{2n}\}$ serve as an orthonormal basis of the normal space $T_p^{\perp }M$, with $F{e}_j=(sin\theta )e_{n+j}$, $\forall j=1,\dots ,n$.

On a Kählerian slant submanifold, the components $h_{ij}^r$ of the second fundamental form h are symmetric, i.e.,

$$h_{ij}^k=h_{jk}^i=h_{ik}^j,\forall i,j,k=1,\dots ,n,$$

where $h_{ij}^k=g(h(e_i,e_j),J{e}_k)$.

In this case, the Gauss equation implies

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =[n(n-1)+3n{cos}^2\theta ]{\displaystyle \frac{c}{4}}\hfill \\ \hfill & -{\parallel h\parallel }^2+n^2{\parallel H\parallel }^2-(n-1)\lambda -n(n-1)\omega \left(H\right).\hfill \end{array}$$

(49)

Also, one can obtain

$$\begin{array}{cc}\hfill n^2{\parallel H\parallel }^2& =\sum _{i}g(h(e_i,e_i),h(e_i,e_i))+\sum _{i\ne j}g(h(e_i,e_i),h(e_j,e_j))\hfill \\ \hfill & =\sum _{i=1}^n[\sum _{j=1}^n{\left(h_{jj}^i\right)}^2+2\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i].\hfill \end{array}$$

(50)

Next, taking into account Equations (49) and (50), we obtain

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =[n(n-1)+3n{cos}^2\theta ]{\displaystyle \frac{c}{4}}\hfill \\ \hfill & -(n-1)\lambda -n(n-1)\omega \left(H\right)\hfill \\ \hfill & +2\sum _i\sum _{j<k}{h}_{jj}^i{h}_{kk}^i-2\sum _{i\ne j}{\left(h_{jj}^i\right)}^2-6\sum _{i<j<k}{\left(h_{ij}^k\right)}^2.\hfill \end{array}$$

(51)

We denote $m={\displaystyle \frac{n+2}{n-1}}$ with $n\ge 2$. We will use the technique in [15].

Then,

$$\begin{array}{cc}\hfill n^2{\parallel H\parallel }^2& -m\{2\tau -n(n-1){\displaystyle \frac{c}{4}}-3n{cos}^2\theta {\displaystyle \frac{c}{4}}+(n-1)\lambda +n(n-1)\omega \left(H\right)\}\hfill \\ \hfill & =\sum _i{\left(h_{ii}^i\right)}^2+(1+2m)\sum _{1\le i\ne j\le }{\left(h_{jj}^i\right)}^2+6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2\hfill \\ \hfill & -2(m-1)\sum _{i=1}^n\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i\hfill \\ \hfill & =\sum _{i=1}^n{\left(h_{ii}^i\right)}^2+6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2+(m-1)\sum _{i=1}^n\sum _{1\le j<k\le n}{(h_{jj}^i-h_{kk}^i)}^2\hfill \\ \hfill & +\left\{1+2m-(n-2)(m-1)\right\}\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2-2(m-1)\sum _{1\le i\ne j\le n}{h}_{ii}^i{h}_{jj}^i\hfill \\ \hfill & =6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2+(m-1)\sum _{i\ne j,k}\sum _{1\le j<k\le n}{(h_{jj}^i-h_{kk}^i)}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{n-1}}\sum _{1\le i\ne j\le n}{\left\{h_{ii}^i-(n-1)(m-1)h_{jj}^i\right\}}^2\ge 0.\hfill \end{array}$$

(52)

So, we can obtain

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge {\displaystyle \frac{2(n+2)}{n^2(n-1)}}\tau -{\displaystyle \frac{n+2}{n}}{\displaystyle \frac{c}{4}}\hfill \\ \hfill & +3{\displaystyle \frac{n+2}{n(n-1)}}{\displaystyle \frac{c}{4}}{cos}^2\theta \hfill \\ \hfill & +{\displaystyle \frac{n+2}{n^2}}\lambda +{\displaystyle \frac{n+2}{n}}\omega \left(H\right)].\hfill \end{array}$$

(53)

□

Next, let us provide an example of a Kählerian slant surface in ${\mathbb{R}}^4$ with the standard Kählerian structure, admitting a semi-symmetric non-metric connection.

Example 2. 

The standard Kählerian structure $J_0$ on ${\mathbb{R}}^4$ is expressed by

$$J_0(x,y,z,w)=(-z,-w,x,y).$$

We consider a semi-symmetric non-metric connection

$${\nabla }_{X}Y={\stackrel{\circ }{\nabla }}_{X}Y+\omega \left(Y\right)X,$$

where $\omega \left(Y\right)=<Y,J_0{e}_1>=\mathsf{\Omega }(e_1,Y)$, Ω is the fundamental 2-form.

It is known that the holomorphic sectional curvature vanishes identically on ${\mathbb{R}}^4$.

We define a two-dimensional Kählerian slant submanifold in ${\mathbb{R}}^4$ with the usual Kählerian structure, admitting the above semi-symmetric non-metric connection, using the following equation:

$$x(u,v)=\left(\right(u+v),kcosv,v-u,ksinv),k>0.$$

An orthonormal frame on $TM$ is given by

$$\begin{array}{cc}\hfill e_1& ={\displaystyle \frac{1}{\sqrt{2}}}(1,0,-1,0),\hfill \\ \hfill e_2& ={\displaystyle \frac{1}{\sqrt{k^2+2}}}(1,-ksinv,1,kcosv),\hfill \end{array}$$

and an orthornormal frame on $T^{\perp }M$ is given by

$$\begin{array}{cc}\hfill e_3=e_{1^{*}}& ={\displaystyle \frac{1}{sin\theta }}F{e}_1.\hfill \\ \hfill e_4=e_{2^{*}}& ={\displaystyle \frac{1}{sin\theta }}F{e}_2.\hfill \end{array}$$

We have $J_0{e}_1={\displaystyle \frac{1}{\sqrt{2}}}(1,0,1,0)$. Then,

$$<J_0{e}_1,e_2>=\sqrt{{\displaystyle \frac{2}{2+k^2}}}.$$

Thus, M is a Kählerian slant surface with slant angle $\theta ={cos}^{-1}\sqrt{{\displaystyle \frac{2}{2+k^2}}}$.

Computing the second fundamental form, we can obtain

$$h(e_1,e_1)=h(e_1,e_2)=0$$

and

$$h(e_2,e_2)={\displaystyle \frac{k}{k^2+2}}{e}_4={\displaystyle \frac{k}{k^2+2}}(0,-cosv,0,-sinv,0).$$

Also, $H={\displaystyle \frac{1}{2}}h(e_2,e_2)\ne 0$. Then, M is not a minimal surface.

In the Gauss equation, we use $X=W=e_1$, $Y=Z=e_2$. Then,

$$R(e_1,e_2,e_2,e_1)=-s(e_2,e_2)+g(h(e_1,e_1),h(e_2,e_2))-g(h(e_1,e_2),h(e_1,e_2))-g(P^{\perp },h(e_2,e_2)).$$

In our case, $s(e_2,e_2)=-{\displaystyle \frac{2}{2+k^2}}$. Then, $R(e_1,e_2,e_2,e_1)=-{\displaystyle \frac{2}{2+k^2}}$.

Similarly, $R(e_2,e_1,e_1,e_2)=0$.

Consequently, $\tau =K\left(\pi \right)=-{\displaystyle \frac{1}{2+k^2}}$.

An improved Euler inequality for Lagrangian submanifolds in complex space forms was established by B.Y. Chen [27].

We state the corresponding result for Lagrangian submanifolds in complex space forms admitting a semi-symmetric non-metric connection.

Corollary 8. 

Let M be an n-dimensional ($n\ge 2$) Lagrangian submanifold of a complex space form $\overline{M}\left(c\right)$ admitting a semi-symmetric non-metric connection. Then,

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge {\displaystyle \frac{2(n+2)}{n^2(n-1)}}\tau -{\displaystyle \frac{n+2}{n}}{\displaystyle \frac{c}{4}}\hfill \\ \hfill & +{\displaystyle \frac{n+2}{n^2}}\lambda +{\displaystyle \frac{n+2}{n}}\omega \left(H\right).\hfill \end{array}$$

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6. Conclusions

In this article, we dealt with the study of submanifolds in complex space forms which admit a semi-symmetric, non-metric connection. This is the first article studying such submanifolds because a sectional curvature for such a connection cannot be defined using the standard definition. We used the sectional curvature recently defined by A. Mihai and the first author in [6].

We proved the Chen–Ricci inequality and Chen’s first inequality for arbitrary submanifolds in a complex space form admitting a semi-symmetric, non-metric connection. We also discussed particular cases of such submanifolds. Moreover, we obtained a generalized Euler inequality for Kählerian slant submanifolds in such ambient spaces.

We point out that corresponding problems in Sasakian settings were investigated by the first author and his coworkers (see [28]).

This study could be continued. One could establish other Chen inequalities and our results could be improved for special submanifolds in complex space forms or in other ambient spaces admitting non-metric connections; for example, in locally metallic product space forms (see [29]).

Similar problems can be considered in the Lorentzian setting.