Intrinsic and Extrinsic Geometry of Kählerian Slant Submanifolds in Complex Space Forms

Abstract

The study of complex space forms plays a central role in differential geometry, as these manifolds provide a natural framework for exploring geometric structures endowed with rich symmetries, extending both Riemannian and Kählerian geometries. In this paper, we extend the definition of Kählerian slant submanifolds when the ambient complex space form admits a quarter-symmetric connection. Furthermore, we establish sharp relationships between intrinsic and extrinsic geometric invariants of Kählerian slant submanifolds in complex space forms admitting a quarter-symmetric connection.

1. Introduction

The concept of semi-symmetric connections was first introduced by Friedmann and Schouten [1], marking an important development in affine differential geometry. Later, Hayden [2] independently considered similar notions. The geometric properties were further examined by Yano [3] in Riemannian manifolds. His work resulted in the formulation of semi-symmetric metric connections.

In 1975, Golab [4] extended this idea and introduced the notion of a quarter-symmetric connection, which generalizes both semi-symmetric metric and non-metric connections. By preserving metric compatibility while permitting a specific torsion component, this connection offers a richer framework for geometric analysis than the classical Levi-Civita connection. Subsequent contributions by Agashe and Chafle [5,6] expanded the theory to include non-metric cases and explored its applications in the geometry of submanifolds.

Addressing Chen’s open problem, which concerns identifying optimal relationships between the intrinsic and extrinsic invariants of Riemannian submanifolds, central challenge in submanifold geometry, Chen introduced the so-called $\delta$-invariants (or Chen invariants) in [7,8]. These invariants provide a powerful framework for formulating and proving sharp inequalities that relate the curvature quantities of the submanifolds.

Following Chen’s pioneering work, numerous researchers have investigated these invariants in various ambient spaces, and manifolds admitting semi-symmetric metric connections, where the definition of sectional curvature still preserves its skew-symmetric properties.

Several B.-Y. Chen inequalities for submanifolds of generalized complex space forms were established in [9]. A Chen inequality for submanifolds of S-space forms tangent to the structure vector fields, with applications to slant immersions, was obtained in [10]. Sharp inequalities involving $\delta$-invariants for submanifolds of generalized complex and generalized Sasakian space forms with arbitrary codimension were studied in [11], while a B.-Y. Chen inequality for semi-slant submanifolds of T-space forms was derived in [12].

Chen inequalities relating intrinsic and extrinsic curvatures were further established for submanifolds of real, complex, and Sasakian space forms endowed with a semi-symmetric metric connection [13,14], and for submanifolds of Riemannian manifolds of nearly quasi-constant curvature equipped with a semi-symmetric non-metric connection [15].

Recently, in [16], the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms were investigated, and the first Chen inequality was established.

When the ambient manifold is endowed with a semi-symmetric non-metric connection, the classical notion of sectional curvature, defined via the symmetric and metric-preserving Levi-Civita connection, becomes inadequate. To overcome this limitation, A. Mihai and the third author proposed in [17] a modified definition of sectional curvature that is compatible with semi-symmetric non-metric connections. This new formulation preserves the essential skew-symmetric properties of curvature and has been successfully applied to the study of various geometric inequalities.

Furthermore, motivated by Golab’s introduction of quarter-symmetric connections [4], several researchers have studied submanifolds in ambient manifolds equipped with this more general type of connection. In particular, Wang [18] developed a unified framework for Chen-type inequalities in complex space forms and Sasakian space forms endowed with quarter-symmetric metric connections. By deriving a generalized Gauss equation adapted to quarter-symmetric connections, Wang established sharp inequalities relating intrinsic invariants—such as Ricci and scalar curvature—to extrinsic quantities defined via the second fundamental form. His results demonstrate that the quarter-symmetric setting simultaneously extends both the semi-symmetric metric and semi-symmetric non-metric cases, thereby providing a broader geometric structure for studying curvature inequalities.

2. Preliminaries

Let $\overline{M}$ be an $(n+k)$-dimensional Riemannian manifold endowed with a Riemannian metric g and its associated Levi-Civita connection $\widehat{\overline{\nabla }}$. We define a linear connection $\overline{\nabla }$ by

$${\overline{\nabla }}_{X}Y={\widehat{\overline{\nabla }}}_{X}Y+{\lambda }_1\omega \left(Y\right)X-{\lambda }_{2}g(X,Y)P,$$

(1)

where P represents a vector field in M defined by $\omega \left(X\right)=g(P,X)$.

When ${\lambda }_1={\lambda }_2=1$, the connection $\overline{\nabla }$ reduces to a semi-symmetric metric connection, whereas for ${\lambda }_1=1$, ${\lambda }_2=0$, it defines a semi-symmetric non-metric connection.

Let

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

denote the curvature tensor of $\overline{\nabla }$. The curvature tensor corresponding to $\widehat{\overline{\nabla }}$, denoted by $\widehat{\overline{R}}$, is expressed analogously.

Define the $(0,2)$-tensors:

$$\begin{array}{cc}\hfill \alpha (X,Y)& =({\widehat{\overline{\nabla }}}_X\omega )\left(Y\right)-{\lambda }_1\omega \left(X\right)\omega \left(Y\right)+{\displaystyle \frac{{\lambda }_2}{2}}g(X,Y)\omega \left(P\right),\hfill \\ \hfill \beta (X,Y)& ={\displaystyle \frac{\omega \left(P\right)}{2}}g(X,Y)+\omega \left(X\right)\omega \left(Y\right).\hfill \end{array}$$

Then, by [18], we obtain

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)& =\widehat{\overline{R}}(X,Y,Z,W)+{\lambda }_1\alpha (X,Z)g(Y,W)-{\lambda }_1\alpha (Y,Z)g(X,W)\hfill \\ \hfill & +{\lambda }_{2}g(X,Z)\alpha (Y,W)-{\lambda }_{2}g(Y,Z)\alpha (X,W)\hfill \\ \hfill & +{\lambda }_2({\lambda }_1-{\lambda }_2)g(X,Z)\beta (Y,W)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)g(Y,Z)\beta (X,W).\hfill \end{array}$$

(2)

Let M be an n-dimensional submanifold of an $(n+k)$-dimensional Riemannian manifold $\overline{M}$. On the submanifold M, we consider the induced quarter-symmetric connection denoted by ∇ and the induced Levi-Civita connection denoted by $\widehat{\nabla }$. Let R and $\widehat{R}$ denote the curvature tensors corresponding to ∇ and $\widehat{\nabla }$, respectively. The vector field P on M can be uniquely decomposed into its tangent and normal components, $P^{\top }$ and $P^{\perp }$, so that $P=P^{\top }+P^{\perp }$.

The Gauss formulas corresponding to the connections ∇ and $\widehat{\nabla }$ can be expressed as

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{X}Y& ={\nabla }_{X}Y+h(X,Y),X,Y\in \mathrm{\Gamma }\left(TM\right),\hfill \\ \hfill {\widehat{\overline{\nabla }}}_{X}Y& ={\widehat{\nabla }}_{X}Y+\widehat{h}(X,Y),X,Y\in \mathrm{\Gamma }\left(TM\right),\hfill \end{array}$$

where $\widehat{h}$ denotes the second fundamental form of M in $\overline{M}$. Moreover, $\widehat{h}$ is related to h via

$$\widehat{h}(X,Y)=h(X,Y)+{\lambda }_{2}g(X,Y)P^{\perp }.$$

(3)

We choose a local orthonormal frame $e_1,\dots ,e_n,e_{n+1},\dots ,e_{n+k}$ on $\overline{M}$ with the property that $e_1,\dots ,e_n$ are tangent to M when restricted to M.

We denote ${\widehat{h}}_{ij}^r=g(\widehat{h}(e_i,e_j),e_r)$, for all $i,j=1,...,n$, $r=n+1,...,n+k$. The squared norm of the second fundamental form h is given by

$${\left|h\right|}^2=\sum _{i,j=1}^{n}g\left(h(e_i,e_j),h(e_i,e_j)\right),$$

and the mean curvature vector of M with respect to ∇ is

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

In the same way, the mean curvature vector with respect to $\widehat{\nabla }$ is

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

By Wang [18], the Gauss equation becomes

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)& =R(X,Y,Z,W)-g\left(h\right(X,W),h(Y,Z\left)\right)+g\left(h\right(Y,W),h(X,Z\left)\right)\hfill \\ \hfill & +({\lambda }_1-{\lambda }_2)g(h(Y,Z),P)g(X,W)\hfill \\ \hfill & +({\lambda }_2-{\lambda }_1)g(h(X,Z),P)g(Y,W).\hfill \end{array}$$

(4)

Let $\pi \subset T_{p}M$ be a 2-plane section at a point $p\in M$, with $\{e_1,e_2\}$ forming an orthonormal basis of $\pi$. When the ambient manifold is equipped with a non-Levi-Civita connection, such as a quarter-symmetric connection or a semi-symmetric non-metric connection, the classical definition of sectional curvature is no longer directly applicable due to the lack of symmetry in the curvature tensor. Specifically, we generally have the following.

$$R(X,Y,Z,W)\ne -R(X,Y,W,Z).$$

To address this, several alternative definitions have been proposed in the literature. For example, Wang [18] proposed a consistent approach for defining sectional curvature in manifolds admitting quarter-symmetric connections, while the third author and A. Mihai [17] proposed a similar framework for manifolds endowed with semi-symmetric non-metric connections.

Motivated by these works, we adopt the following symmetrized expression to define the sectional curvature $K\left(\pi \right)$ associated with a non-metric or torsion-affected connection.

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

(5)

This definition ensures that $K\left(\pi \right)$ remains real-valued and well-defined, even when the curvature tensor lacks the usual anti-symmetry in the last two arguments.

For any orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{p}M$, the scalar curvature $\tau$ at p is defined by

$$\tau \left(p\right)=\sum _{1\le i<j\le n}K(e_i\wedge e_j)={\displaystyle \frac{1}{2}}\sum _{1\le i,j\le n}R(e_i,e_j,e_j,e_i).$$

(6)

The study of Riemannian, complex, almost contact, hypercomplex, quaternionic, and Cauchy–Riemann structures falls within the broad framework of G-structures. This area plays a fundamental role in modern differential geometry, global analysis, and mathematical physics.

In this subsection, we provide a brief overview of the basic notions and properties related to the differential geometry of manifolds endowed with complex structures. For further details, the reader is referred to [19].

A complex manifold $(\overline{M},g)$ of real dimension $2m$ is called a Hermitian manifold if there exists an almost complex structure J, tensor field of type $(1,1)$, satisfying

$$J^{2}X=-X,$$

such that

$$g(JX,JY)=g(X,Y),$$

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

If the almost complex structure J is parallel with respect to the Levi-Civita connection $\widehat{\overline{\nabla }}$, then the manifold $\overline{M}$ is Kähler.

A 2-plane section invariant by J is called a holomorphic plane section. A complex space form is a Kähler manifold whose holomorphic sectional curvature is constant, equal to c. It is known that complex space forms have the highest degree of homogeneity because their isometries spaces have the maximum dimension.

For a complex space form $\overline{M}\left(c\right)$, the curvature tensor satisfies (see, for instance, [20])

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

(7)

for all vector fields $X,Y,Z$ on $\overline{M}\left(c\right)$.

For a $2m$-dimensional complex space form $\overline{M}\left(c\right)$ with constant holomorphic sectional curvature c and a quarter-symmetric connection $\tilde{\nabla }$, the curvature tensor $\overline{R}$ is determined from (2) and (7) as follows:

$$\begin{array}{cc}\hfill & \overline{R}(X,Y,Z,W)={\displaystyle \frac{c}{4}}[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\hfill \\ \hfill & +g(Z,JY)g(JX,W)-g(Z,JX)g(JY,W)+2g(X,JY)g(JZ,W)]\hfill \\ \hfill & +{\lambda }_1\alpha (X,Z)g(Y,W)-{\lambda }_1\alpha (Y,Z)g(X,W)\hfill \\ \hfill & +{\lambda }_{2}g(X,Z)\alpha (Y,W)-{\lambda }_{2}g(Y,Z)\alpha (X,W)\hfill \\ \hfill & +{\lambda }_2({\lambda }_1-{\lambda }_2)g(X,Z)\beta (Y,W)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)g(Y,Z)\beta (X,W).\hfill \end{array}$$

(8)

Let $x:M⟶\overline{M}$ be an isometric immersion of an n-dimensional Riemannian manifold M in a $2m$-dimensional Hermitian manifold $\overline{M}$, equipped with an almost complex structure J and a compatible Riemannian metric g.

For any tangent vector field $X\in \mathrm{\Gamma }\left(TM\right)$, the image $JX$ can be decomposed into tangential and normal components as

$$JX=TX+FX,$$

where $TX$ is tangent to M and $FX$ is normal to M. Thus, T acts as an endomorphism of the tangent bundle $TM$, while F defines a 1-form on $TM$ taking values in the normal bundle $T^{\perp }M$.

For a nonzero vector X tangent to M at a point $p\in M$, denote by $\theta \left(X\right)$ the angle between $JX$ and $T_{p}M$. This angle $\theta \left(X\right)$ is termed the Wirtinger angle of X.

The immersion $x:M\to \overline{M}$ is said to be a slant immersion if $\theta \left(X\right)$ is constant, that is, it is independent of both $p\in M$ and the choice of the tangent vector $X\in T_{p}M$. The common value of $\theta \left(X\right)$ is termed the slant angle of the immersion.

• When $\theta =0$, the immersion is invariant (and it becomes complex because $\overline{M}$ is Hermitian);
• When $\theta ={\displaystyle \frac{\pi }{2}}$, the immersion is totally real (also called anti-invariant);
• For $0<\theta <{\displaystyle \frac{\pi }{2}}$, the immersion is called a proper slant immersion.

For slant submanifolds, the endomorphism T satisfies the algebraic condition $T^2=\epsilon I$, for some constant $\epsilon \in [-1,0]$, where I denotes the identity transformation on $TM$. Accordingly,

$$\epsilon =\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}\theta =0,\left(\mathrm{invariant}\mathrm{case}\right),\hfill \\ 0,\hfill & \mathrm{if}\theta =\frac{\pi }{2},\left(\mathrm{totally}\mathrm{real}\mathrm{case}\right),\hfill \\ -{\cos }^2\theta ,\hfill & \mathrm{if}0<\theta <\frac{\pi }{2},\left(\mathrm{proper}\mathrm{slant}\mathrm{case}\right).\hfill \end{array}\right.$$

A proper slant submanifold is called a Kählerian slant submanifold when its canonical endomorphism T is parallel with respect to the Levi-Civita connection $\widehat{\nabla }$ of M, that is, $\widehat{\nabla }T=0.$

It is known [21] that this condition is equivalent to

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

(9)

where $A_N$ denotes the shape operator corresponding to the normal vector field N.

If $\{e_1,\dots ,e_n\}$ is an orthonormal basis of $T_{p}M$ at $p\in M$, then we set

$$e_i^{*}={\displaystyle \frac{1}{\sin \theta }}F{e}_i,i=1,\dots ,n.$$

We denote ${\widehat{h}}_{ij}^k=g(\widehat{h}(e_i,e_j),e_k^{*}),\forall i,j,k=1,\dots ,n$. Because the submanifold M is a Kählerian slant submanifold, Equation (9) implies that the components of the second fundamental form are symmetric, that is,

$${\widehat{h}}_{ij}^k={\widehat{h}}_{ik}^j={\widehat{h}}_{jk}^i,\forall i,j,k=1,\dots ,n.$$

In this situation, M itself inherits a Kähler structure; it becomes a Kähler manifold with respect to the induced metric and the almost complex structure

$$J^{\prime }=\left(\sec \theta \right)T,$$

where $\theta$ denotes the slant angle.

Explicit constructions and examples of proper slant and Kählerian slant submanifolds can be found in classical references (see, for instance, Chen [21]).

Lemma 1. 

Let $f_1(x_1,x_2,...,x_n)$ be a function of ${\mathbb{R}}^n$ defined by

$$f_1(x_1,x_2,...,x_n)=x_1\sum _{j=2}^n{x}_j-\sum _{j=2}^n{x}_j^2.$$

If $x_1+x_2+...+x_n=2na$, then we have

$$f_1(x_1,x_2,...,x_n)\le {\displaystyle \frac{n-1}{4n}}{(x_1+x_2+...+x_n)}^2,$$

with the equality sign holding if and only if ${\displaystyle \frac{1}{n+1}}{x}_1=x_2=...=x_n=a$.

Lemma 2. 

Let $f_2(x_1,x_2,...,x_n)$ be a function on ${\mathbb{R}}^n$ defined by

$$f_2(x_1,x_2,...,x_n)=x_1\sum _{j=2}^n{x}_j-x_1^2.$$

If $x_1+x_2+...+x_n=4a$, then we have

$$f_2(x_1,x_2,...,x_n)\le {\displaystyle \frac{1}{8}}{(x_1+x_2+...+x_n)}^2,$$

with the equality sign holding if and only if $x_1=a$ and $x_2+...+x_n=3a$.

3. First Chen Like Inequality

In this section, we derive a first Chen-type inequality for Kählerian slant submanifolds of a complex space form equipped with a quarter-metric connection.

Theorem 1. 

Let M be an $n=2s$-dimensional $(n\ge 3)$ Kählerian slant submanifold isometrically immersed in a $2n$-dimensional complex space form $\overline{M}\left(c\right)$ endowed with quarter-symmetric connection $\overline{\nabla }$. We assume that the vector field P is tangent to M. Then, for any $p\in M$ and any 2-plane section $\pi \subset T_{p}M$, we have

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le {\displaystyle \frac{2n-3}{2(2n+3)}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right),\hfill \end{array}$$

(10)

where ${\mathsf{\Theta }}^2\left(\pi \right)=g^2(J{e}_1,e_2)$, for any orthonormal vectors $e_1,e_2\in \pi$.

The equality at a point $p\in M$ holds if and only if, with respect to a suitable orthonormal basis $\{e_1,...,e_n\}$, the second fundamental form has the following expressions:

$$\begin{array}{cccc}\hfill & h(e_1,e_1)=aF{e}_1+3bF{e}_3,\hfill & \hfill h(e_1,e_3)=3bF{e}_1,& h(e_3,e_j)=4bF{e}_j,\hfill \\ \hfill & h(e_2,e_2)=-aFe1+3bF{e}_3,\hfill & \hfill h(e_2,e_3)=3bF{e}_2,& h(e_j,e_k)=4bF{e}_3{\delta }_{jk},\hfill \\ \hfill & h(e_1,e_2)=-aF{e}_2,\hfill & \hfill h(e_3,e_3)=12bF{e}_3,& h(e_1,e_j)=h(e_2,e_j)=0,\hfill \end{array}$$

for some numbers a and b, and $j,k\in \{4,\dots ,n\}$.

Proof. 

Let $p\in M$ and let $\pi \subset T_{p}M$ be a 2-plane section. We choose an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{p}M$ such that $e_1,e_2\in \pi$ and an orthonormal basis $\{e_{n+1},\dots ,e_{4s}\}$ of the normal space $T_p^{\perp }M$.

Since $\overline{M}\left(c\right)$ is a complex space form equipped with a quarter-symmetric connection, it follows from the amended Gauss Equation (4) that

$$\begin{array}{cc}\hfill 2\tau =& \sum _{i,j=1}^{n}R(e_i,e_j,e_j,e_i)=\sum _{i,j=1}^n[\overline{R}(e_i,e_j,e_j,e_i)-g(h(e_i,e_j),h(e_j,e_i))\hfill \\ \hfill & +g(h(e_i,e_i),h(e_j,e_j))-({\lambda }_1-{\lambda }_2)g(h(e_j,e_j),P)g(e_i,e_i)\hfill \\ \hfill & -({\lambda }_2-{\lambda }_1)g(h(e_i,e_j),P)g(e_j,e_i)].\hfill \end{array}$$

(11)

Using the Formula (8) for the curvature tensor, we calculate $\overline{R}(e_i,e_j,e_j,e_i)$ by taking $X=W=e_i$ and $Y=Z=e_j$, for all $i,j=1,\dots ,2s$.

$$\begin{array}{cc}\hfill \overline{R}(e_i,e_j,e_j,e_i)& ={\displaystyle \frac{c}{4}}\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\hfill \\ \hfill & +[g(e_j,J{e}_j)g(J{e}_i,e_i)\hfill \\ \hfill & -g(e_j,J{e}_i)g(J{e}_j,e_i)+2g(e_i,J{e}_j)g(J{e}_j,e_i)\left]\right\}\hfill \\ \hfill & +{\lambda }_1\alpha (e_i,e_j)g(e_j,e_i)-{\lambda }_1\alpha (e_j,e_j)g(e_i,e_i)\hfill \\ \hfill & +{\lambda }_{2}g(e_i,e_j)\alpha (e_j,e_i)-{\lambda }_{2}g(e_j,e_j)\alpha (e_i,e_i)\hfill \\ \hfill & +{\lambda }_2({\lambda }_1-{\lambda }_2)g(e_i,e_j)\beta (e_j,e_i)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)g(e_j,e_j)\beta (e_i,e_i).\hfill \end{array}$$

(12)

This equation yields

$$\begin{array}{cc}\hfill \sum _{i,j=1}^n\overline{R}(e_i,e_j,e_j,e_i)& ={\displaystyle \frac{c}{4}}[n(n-1)+3\sum _{i,j=1}^n{g}^2(T{e}_i,e_j)]\hfill \\ \hfill & -({\lambda }_1+{\lambda }_2)(n-1)\sum _{i=1}^n\alpha (e_i,e_i)\hfill \\ & -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\sum _{i=1}^n\beta (e_i,e_i).\hfill \end{array}$$

(13)

We denote $\mathrm{tr}\left(\alpha \right)=\lambda$ and $\mathrm{tr}\left(\beta \right)=\mu$, and ${\sum }_{i,j=1}^n{g}^2(T{e}_i,e_j)={\parallel T\parallel }^2$. We can rewrite (13) as

$$\begin{array}{cc}\hfill \sum _{i,j=1}^n\overline{R}(e_i,e_j,e_j,e_i)& ={\displaystyle \frac{c}{4}}\left[n(n-1)+{3\parallel T\parallel }^2\right]\hfill \\ \hfill & -({\lambda }_1+{\lambda }_2)(n-1)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu .\hfill \end{array}$$

(14)

Now, substituting (14) in (11), we get

$$\begin{array}{cc}\hfill 2\tau & ={\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & -({\lambda }_1+{\lambda }_2)(n-1)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu \hfill \\ \hfill & +\sum _{r=n+1}^{4s}\sum _{1\le i,j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2].\hfill \end{array}$$

(15)

Let $\pi$ be the plane spanned by $e_1,e_2$. Then, by the Gauss equation, taking $X=W=e_1$ and $Y=Z=e_2$, we obtain

$$\begin{array}{c}\hfill R(e_1,e_2,e_2,e_1)=\overline{R}(e_1,e_2,e_2,e_1)-g(h(e_1,e_2),h(e_2,e_1))+g(h(e_1,e_1),h(e_2,e_2)).\end{array}$$

(16)

Similarly, by substituting $X=W=e_1$ and $Y=Z=e_2$ into the formula for constant holomorphic sectional curvature, we have

$$\begin{array}{cc}\hfill \overline{R}(e_1,e_2,e_2,e_1)& ={\displaystyle \frac{c}{4}}\{g(e_2,e_2)g(e_1,e_1)-g(e_1,e_2)g(e_2,e_1)\hfill \\ \hfill & +[g(e_2,J{e}_2)g(J{e}_1,e_1)\hfill \\ \hfill & -g(e_2,J{e}_1)g(J{e}_2,e_1)+2g(e_1,J{e}_2)g(J{e}_2,e_1)\left]\right\}\hfill \\ \hfill & +{\lambda }_1\alpha (e_1,e_2)g(e_2,e_1)-{\lambda }_1\alpha (e_2,e_2)g(e_1,e_1)\hfill \\ \hfill & +{\lambda }_{2}g(e_1,e_2)\alpha (e_2,e_1)-{\lambda }_{2}g(e_2,e_2)\alpha (e_1,e_1)\hfill \\ \hfill & +{\lambda }_2({\lambda }_1-{\lambda }_2)g(e_1,e_2)\beta (e_2,e_1)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)g(e_2,e_2)\beta (e_1,e_1),\hfill \end{array}$$

(17)

which simplifies to

$$\begin{array}{cc}\hfill \overline{R}(e_1,e_2,e_2,e_1)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_2)]-{\lambda }_1\alpha (e_2,e_2)-{\lambda }_2\alpha (e_1,e_1)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)\beta (e_1,e_1).\hfill \end{array}$$

(18)

Substituting (18) in (16) yields

$$\begin{array}{cc}\hfill R(e_1,e_2,e_2,e_1)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_2)]-{\lambda }_1\alpha (e_2,e_2)-{\lambda }_2\alpha (e_1,e_1)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)\beta (e_1,e_1)+\sum _{r=n+1}^{2n}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

(19)

Also, by taking $X=W=e_2$ and $Y=Z=e_1$ in the Gauss equation, we obtain

$$\begin{array}{cc}\hfill R(e_2,e_1,e_1,e_2)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_2)]-{\lambda }_1\alpha (e_1,e_1)-{\lambda }_2\alpha (e_2,e_2)\hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)\beta (e_2,e_2)+\sum _{r=n+1}^{2n}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

(20)

Then by the formula of the sectional curvature

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

we have

$$\begin{array}{cc}\hfill K\left(\pi \right)& ={\displaystyle \frac{c}{4}}+3{\displaystyle \frac{c}{4}}{g}^2(J{e}_1,e_2)\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right)\hfill \\ \hfill & +\sum _{r=n+1}^{2n}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

(21)

For a plane section $\pi =$ span$\{e_1,e_2\}$, we denote ${\mathsf{\Theta }}^2\left(\pi \right)=g^2(J{e}_1,e_2)$. Then

$$\begin{array}{cc}\hfill K\left(\pi \right)& ={\displaystyle \frac{c}{4}}[1+3{\mathsf{\Theta }}^2\left(\pi \right)]-{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right)+\sum _{r=n+1}^{2n}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

(22)

Now from (15),

$$\begin{array}{cc}\hfill \tau & ={\displaystyle \frac{c}{8}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +\sum _{r=n+1}^{4s}\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2],\hfill \end{array}$$

(23)

and, in addition, using Equations (23) and (22), we have

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& ={\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right)\hfill \\ \hfill & +\sum _{r=n+1}^{4s}\sum _{1\le i<j\le n}\left[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2\right]-\sum _{r=n+1}^{4s}\left(h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2\right),\hfill \end{array}$$

(24)

or equivalently,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& ={\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right)\hfill \\ \hfill & +\sum _{r=n+1}^{2n}\{\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n{\left(h_{1j}^r\right)}^2-\sum _{2\le i<j\le n}{\left(h_{ij}^r\right)}^2\}.\hfill \end{array}$$

(25)

It follows that

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le {\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right)\hfill \\ \hfill & +\sum _{r=n+1}^{2n}\{\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n{\left(h_{1j}^r\right)}^2-\sum _{j=3}^n{\left(h_{jj}^{n+1}\right)}^2-\sum _{i,j\in \{2,3,...,n\},i\ne j}{\left(h_{jj}^{n+i}\right)}^2\}.\hfill \end{array}$$

(26)

For any $i=1,...,n$, we denote $e_i^{*}={\displaystyle \frac{1}{\sin \theta }}F{e}_i$. Then $\{e_1^{*},...,e_n^{*}\}$ is an orthonormal basis of $T_p^{\perp }M$.

Simply, we put $h_{ij}^k=g(h(e_i,e_j),e_k^{*})$, $i,j,k=1,\dots ,n$.

Since M is a Kählerian slant submanifold, the components of the second fundamental form are symmetric, that is,

$${\widehat{h}}_{ij}^k={\widehat{h}}_{ik}^j={\widehat{h}}_{jk}^i,\forall i,j,k=1,\dots ,n.$$

We assume that the vector field P is tangent to M. Then, $\widehat{h}=h$.

In our inequality we use $h_{1j}^1=h_{11}^j$, $h_{1j}^j=h_{jj}^1$, for $3\le j\le n$, and $h_{ij}^j=h_{jj}^i$, for $2\le i\ne j\le n$.

The above inequality implies that

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n\{\sum _{j=1}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n{\left(h_{11}^j\right)}^2-\sum _{j=3}^n{\left(h_{jj}^1\right)}^2-\sum _{2\le i\ne j\le n}{\left(h_{jj}^i\right)}^2\}\hfill \\ \hfill & +{\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right).\hfill \end{array}$$

(27)

In order to achieve the proof, we will use ideas from [22]. In this respect, recall the following inequalities

$$\begin{array}{cc}\hfill & \sum _{j=1}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r-\sum _{j=3}^n{\left(h_{jj}^r\right)}^2\hfill \\ \hfill & \le {\displaystyle \frac{n-2}{2(n+1)}}{(h_{11}^r+\dots +h_{nn}^r)}^2\hfill \\ \hfill & \le {\displaystyle \frac{2n-3}{2(2n+3)}}{(h_{11}^r+\dots +h_{nn}^r)}^2\hfill \end{array}$$

(28)

for $r=1,2$. The first inequality in (28) can be rewritten as

$$\begin{array}{c}\hfill \sum _{j=3}^n{(h_{11}^r+h_{22}^r-3h_{jj}^r)}^2+3\sum _{3\le i<j\le n}{(h_{ii}^r-h_{jj}^r)}^2\ge 0.\end{array}$$

The equality holds if and only if

$$3h_{jj}^r=h_{11}^r+h_{22}^r,\forall j=3,\dots ,n.$$

Similarly, the equality in the second inequality holds if and only if

$$h_{11}^r+h_{22}^r=0\mathrm{and}{h}_{jj}^r=0,\forall j=3,\dots ,n,r=1,2.$$

Moreover, we have

$$\begin{array}{cc}\hfill & \sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r-\sum _{j=1,j\ne r}^n\left(h_{jj}^r\right)\hfill \\ \hfill & \le {\displaystyle \frac{2n-3}{2(2n+3)}}{(h_{11}^r+\dots +h_{nn}^r)}^2,\hfill \end{array}$$

(29)

for $r=3,\dots ,n$, which is equivalent to

$$\begin{array}{cc}\hfill & \sum _{j=3}^n{[2(h_{11}^r+h_{22}^r)h_{jj}^r-3h_{jj}^r]}^2+(2n+3){(h_{11}^r-h_{22}^r)}^2\hfill \\ \hfill & +6\sum _{3\le i<j\le n,i,j\ne r}{(h_{ii}^r-h_{jj}^r)}^2+2\sum _{j=3}^n{(h_{rr}^r-h_{jj}^r)}^2\hfill \\ \hfill & +3{[h_{rr}^r-2(h_{11}^r+h_{22}^r)]}^2\ge 0.\hfill \end{array}$$

(30)

The equality holds if and only if

$h_{11}^r=h_{22}^r=3{\lambda }^r,h_{jj}^r=4{\lambda }^r$; $\forall j=3,\dots ,n,j\ne r,r=3,\dots ,n$, and $h_{rr}^r=12{\lambda }^r,{\lambda }^r\in \mathbb{R}.$

We will finish the proof by using the inequalities (28) and (29); it follows that (27) implies

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le {\displaystyle \frac{2n-3}{2(2n+3)}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{c}{8}}[(n-2)(n+1)+3(n{\cos }^2\theta -2{\mathsf{\Theta }}^2\left(\pi \right))]\hfill \\ \hfill & -{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|\pi }\right)+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|\pi }\right),\hfill \end{array}$$

(31)

which is the inequality to be proven.

The equality holds if and only if

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

for some numbers a and b, and $j,k\in \{4,...,n\}$.    □

4. Upper Bound and Lower Bound on Ricci Curvature

The Ricci curvature plays a fundamental role in differential geometry and mathematical physics, as it governs the geometric structure of manifolds and appears naturally in the study of Einstein manifolds and curvature inequalities.

In [8], B.-Y. Chen established a sharp inequality providing a lower bound for the Ricci curvature of any n-dimensional Riemannian submanifold of a real space form $\overline{M}\left(c\right)$ with constant sectional curvature c. Specifically, he proved that

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

which is known as the Chen–Ricci inequality.

The same inequality holds for Lagrangian submanifolds in a complex space form $\overline{M}\left(c\right)$ as well (see [23]).

In [24], the authors extended the Chen–Ricci inequality for submanifolds in complex space forms.

This upper bound on Ricci curvature was improved by A. Mihai and I. Rădulescu [25] for Kählerian slant submanifolds in complex space forms.

In this section, we further extend the relationship between the upper and lower bounds on Ricci curvature to Kählerian slant submanifolds in complex space forms admitting a quarter-symmetric connection.

Theorem 2. 

Let $\overline{M}\left(c\right)$ be a $2n$-dimensional complex space form admitting a quarter-symmetric connection and M an n-dimensional Kählerian slant submanifold tangent to p. Then, for any $p\in M$ and any unit vector $X\in T_{p}M$, the Ricci curvature has the following upper bound:

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{c}{4}}(n-1+{\cos }^2\theta )-{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (e_1,e_1)-\mathtt{tr}\alpha ]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (e_1,e_1)+\mathtt{tr}\beta ]+{\displaystyle \frac{n(n-1)}{4}}{\parallel H\parallel }^2.\hfill \end{array}$$

(32)

Moreover, the equality sign of (32) holds identically if and only if either

$\left(i\right)$ $c=0$ and M is totally geodesic, or

$\left(ii\right)$ $n=2$, $c<0$ and M is a slant H-umbilical surface with $\lambda =3\mu$.

Proof. 

At a point $p\in M$, let $X\in T_{p}M$ be a unit vector. We choose an orthonormal basis $\{e_1=X,e_2,\dots ,e_n\}\subset T_{p}M$ and define

$$\left\{e_1^{*}={\displaystyle \frac{1}{\sin \theta }}F{e}_1,\dots ,e_n^{*}={\displaystyle \frac{1}{\sin \theta }}F{e}_n\right\}$$

to obtain an orthonormal basis of $T_p^{\perp }M$.

For any $j=2,\dots ,n$, we compute the sectional curvature $K(e_1\wedge e_j)$ of the plane section spanned by $e_1$ and $e_j$ similarly to (22).

$$\begin{array}{cc}\hfill K(e_1\wedge e_j)& ={\displaystyle \frac{c}{4}}[1+3g^2(J{e}_1,e_j)]-{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}(\alpha (e_1,e_1)+\alpha (e_j,e_j))\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}(\beta (e_1,e_1)+\beta (e_j,e_j))+\sum _{r=n+1}^{2n}[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2].\hfill \end{array}$$

(33)

By summing the above equations for $j=2,\dots ,n$, we obtain

$$\begin{array}{cc}\hfill Ric\left(X\right)& =\sum _{j=2}^{n}K(e_1\wedge e_j)={\displaystyle \frac{c}{4}}(n-1+\parallel T{e}_1{\parallel }^2)-{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (e_1,e_1)+\mathtt{tr}\alpha ]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (e_1,e_1)+\mathtt{tr}\beta ]+\sum _{r=n+1}^{2n}\sum _{j=2}^n[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2].\hfill \end{array}$$

(34)

We assume that P is tangent to M. Then $\widehat{h}=h$. We denote $h_{ij}^k=g(h(e_i,e_j),e_k^{*}),i,j,k=1,\dots ,n$. It follows from Equation (34) that

$$\begin{array}{cc}\hfill Ric\left(X\right)& -{\displaystyle \frac{c}{4}}(n-1+{\cos }^2\theta )+{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (e_1,e_1)+\mathtt{tr}\alpha ]\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (e_1,e_1)+\mathtt{tr}\beta ]\hfill \\ \hfill & \le \sum _{r=1}^n\sum _{j=2}^n[h_{11}^r{h}_{jj}^r-{\left(h_{1j}^r\right)}^2].\hfill \end{array}$$

(35)

Since M is a Kählerian slant submanifold, $h_{1j}^1=h_{11}^j$ and $h_{1j}^1=h_{jj}^1$. It follows that the above inequality gives

$$\begin{array}{cc}\hfill Ric\left(X\right)& -{\displaystyle \frac{c}{4}}(n-1+{\cos }^2\theta )+{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (e_1,e_1)+\mathtt{tr}\alpha ]\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (e_1,e_1)+\mathtt{tr}\beta ]\hfill \\ \hfill & \le \sum _{r=1}^n\sum _{j=2}^n{h}_{11}^r{h}_{jj}^r-\sum _{j=2}^n{\left(h_{11}^j\right)}^2-\sum _{j=2}^n{\left(h_{jj}^1\right)}^2.\hfill \end{array}$$

(36)

Because $n{H}_1=h_{11}^1+h_{22}^1+...+h_{nn}^1$, Lemma 1 implies

$$f_1(h_1^{11},h_{22}^1,...,h_{nn}^1)\le {\displaystyle \frac{n-1}{4n}}{\left(n{H}^1\right)}^2={\displaystyle \frac{n(n-1)}{4n}}{\left(H^1\right)}^2.$$

(37)

We denote $f_r(h_{11}^r,h_{22}^r,...,h_{nn}^r)=h_{11}^r{\sum }_{j=2}^n{h}_{jj}^r-{\left(h_{11}^r\right)}^2$, $r=2,...,n$.

Lemma 2 implies

$$f_r(h_{11}^r,h_{22}^r,...,h_{nn}^r)\le {\displaystyle \frac{1}{8}}{\left(n{H}^r\right)}^2\le {\displaystyle \frac{n(n-1)}{4}}{\left(H^r\right)}^2.$$

(38)

By Equations (36)–(38), we obtain

$$\begin{array}{cc}\hfill Ric\left(X\right)& \le {\displaystyle \frac{c}{4}}(n-1+{\cos }^2\theta )-{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (e_1,e_1)-\mathtt{tr}\alpha ]\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (e_1,e_1)+\mathtt{tr}\beta ]+{\displaystyle \frac{n(n-1)}{4}}{\parallel H\parallel }^2.\hfill \end{array}$$

(39)

For the equality case, we use similar arguments as in [26].    □

F. Malek and L. Nejadakbary in [27] established a lower bound on the Ricci curvature of certain submanifolds.

Inspired by their work, we now derive a reverse Chen–Ricci inequality, which provides a lower on the Ricci curvature of Kählerian slant submanifolds in complex space forms admitting a quarter-symmetric connection.

Corollary 1. 

Let $\overline{M}\left(c\right)$ be a $2n$-dimensional complex space form admitting a quarter-symmetric connection and M an n-dimensional Kählerian slant submanifold tangent to p. Then, for any $p\in M$ and any unit vector $X\in T_{p}M$, the Ricci curvature has the following lower bound:

$$\begin{array}{cc}\hfill Ric\left(X\right)& \ge (n-1)\tau \left(p\right)-{\displaystyle \frac{(n-1)(2n-3)}{2(2n+3)}}{\parallel H\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{c}{8}}[(n-2)(n^2-1)+3(n-2)(n+1){\cos }^2\theta ]\hfill \\ \hfill & +{\displaystyle \frac{({\lambda }_1+{\lambda }_2){(n-1)}^2\lambda }{2}}+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2){(n-1)}^2\mu }{2}}\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (X,X)+\mathtt{tr}\alpha ]\hfill \\ & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (X,X)+\mathtt{tr}\beta ].\hfill \end{array}$$

(40)

Proof. 

From the inequality (10) we get

$$\begin{array}{cc}\hfill K(X\wedge Y)& \ge \tau \left(p\right)-{\displaystyle \frac{2n-3}{2(2n+3)}}{\parallel H\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3\{\parallel T{\parallel }^2-2{\mathsf{\Theta }}^2(X\wedge Y)\}\right]\hfill \\ \hfill & +{\displaystyle \frac{({\lambda }_1+{\lambda }_2)(n-1)\lambda }{2}}+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu }{2}}\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}\mathtt{tr}\left({\alpha }_{|X\wedge Y}\right)-{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}\mathtt{tr}\left({\beta }_{|X\wedge Y}\right).\hfill \end{array}$$

(41)

Putting $Y=e_2,\dots ,e_n$ and summing, we derive

$$\begin{array}{cc}\hfill Ric\left(X\right)& \ge (n-1)\tau \left(p\right)-{\displaystyle \frac{(n-1)(2n-3)}{2(2n+3)}}{\parallel H\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{c}{8}}[(n-2)(n^2-1)+3(n-2)(n+1){\cos }^2\theta ]\hfill \\ \hfill & +{\displaystyle \frac{({\lambda }_1+{\lambda }_2){(n-1)}^2\lambda }{2}}+{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2){(n-1)}^2\mu }{2}}\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_1+{\lambda }_2}{2}}[(n-2)\alpha (X,X)+\mathtt{tr}\alpha ]\hfill \\ & -{\displaystyle \frac{{\lambda }_2({\lambda }_1-{\lambda }_2)}{2}}[(n-2)\beta (X,X)+\mathtt{tr}\beta ].\hfill \end{array}$$

(42)

□

5. Generalized Euler Inequality for Kählerian Slant Submanifolds

In this section, we establish an improved version of the generalized Euler inequality, providing a lower bound for the mean curvature of a Kählerian slant submanifold in a complex Riemannian manifold admitting a quarter-metric connection.

B.-Y. Chen [28] derived the following generalized Euler inequality for a n-dimensional submanifold in a Riemannian space form of constant sectional curvature c:

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

with equality holding identically if and only if the submanifold is totally umbilical.

Theorem 3. 

Let M be an n-dimensional Kählerian slant submanifold isometrically immersed in a $2n$-dimensional complex space form admitting a quarter-symmetric connection with P tangent to M. Then, the mean curvature vector satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge {\displaystyle \frac{2(n+2)}{n^2(n-1)}}\tau \left(p\right)-{\displaystyle \frac{(n+2)}{n^2(n-1)}}{\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & +{\displaystyle \frac{n+2}{n^2}}(({\lambda }_1+{\lambda }_2)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu .\hfill \end{array}$$

(43)

Moreover, equality holds identically if and only if M is an H-umbilical submanifold.

Proof. 

Consider an n-dimensional Kählerian slant submanifold of a $2n$-dimensional complex Riemannian manifolds admitting a quarter-symmetric connection.

Recall Equation (15)

$$\begin{array}{cc}\hfill 2\tau & ={\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & -({\lambda }_1+{\lambda }_2)(n-1)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu \hfill \\ \hfill & +\sum _{r=n+1}^{4s}\sum _{1\le i,j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2].\hfill \end{array}$$

(44)

By the definition of the mean curvature we have

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

(45)

Next, taking into account Equations (44) and (45), we obtain

$$\begin{array}{cc}\hfill 2\tau & ={\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]-({\lambda }_1+{\lambda }_2)(n-1)\lambda \hfill \\ \hfill & -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu \hfill \\ \hfill & +2\sum _i^n\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i-2\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2-6\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2.\hfill \end{array}$$

(46)

The last equation can be expressed as

$$\begin{array}{cc}\hfill 2\tau & ={\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & -(n-1)({\lambda }_1+{\lambda }_2)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu \hfill \\ \hfill & +2\sum _i^n\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i-2\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2-6\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2.\hfill \end{array}$$

(47)

Inspired by Chen’s paper [29], we consider the parameter $m={\displaystyle \frac{n+2}{n-1}}$, with $n\ge 2$. To study the inequality for ${\parallel H\parallel }^2$, we follow the technique used in [29]. Specifically, we obtain

$$\begin{array}{cc}\hfill n^2{\parallel H\parallel }^2& -m\{2\tau \left(p\right)-{\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & +(n-1)[({\lambda }_1+{\lambda }_2)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu ]\}\hfill \\ \hfill & =\sum _{i=1}^n{\left(h_{ii}^i\right)}^2+(1+2m)\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2+6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2\hfill \\ \hfill & -2(m-1)\sum _i^n\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i\hfill \\ \hfill & =\sum _{i=1}^n{\left(h_{ii}^i\right)}^2+6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2+(m-1)\sum _{i=1}^n\sum _{1\le j<k\le n}{(h_{jj}^i-h_{kk}^i)}^2\hfill \\ \hfill & +\left[1+2m-(n-2)(m-1)\right]\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2-2(m-1)\sum _{1\le i\ne j\le n}{h}_{ii}^i{h}_{jj}^i\hfill \\ \hfill & =6m\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2+(m-1)\sum _{i\ne j,k}\sum _{1\le j<k\le n}{(h_{jj}^i-h_{kk}^i)}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{n-1}}\sum _{1\le i\ne j\le n}{\left[h_{ii}^i-(n-1)(m-1)h_{jj}^i\right]}^2\ge 0.\hfill \end{array}$$

(48)

Consequently, we obtain

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge {\displaystyle \frac{2(n+2)}{n^2(n-1)}}\tau \left(p\right)-{\displaystyle \frac{(n+2)}{n^2(n-1)}}{\displaystyle \frac{c}{4}}\left[n(n-1)+3n{\cos }^2\theta \right]\hfill \\ \hfill & +{\displaystyle \frac{n+2}{n^2}}(({\lambda }_1+{\lambda }_2)\lambda -{\lambda }_2({\lambda }_1-{\lambda }_2)(n-1)\mu ).\hfill \end{array}$$

(49)

The equality holds if and only if

$$\begin{array}{cc}\hfill h_{ij}^k=& 0,\forall 1\le i<j<k\le n;\hfill \\ \hfill h_{ii}^i=& 3h_{jj}^i,\forall 1\le i\ne j\le n.\hfill \end{array}$$

This implies that M is an H-umbilical submanifold (see [30]).    □

6. Example

It is well known [21] that any proper slant surface in a Kähler manifold is Kählerian slant.

We give an example of a 4-dimensional Kählerian slant submanifold of $C^4$.

For any $k>0$,

$$x(u,v,w,z)=(u,v,k\sin w,k\sin z,kw,kz,k\cos w,k\cos z)$$

defines a Kählerian slant submanifold in $C^4$ with slant angle $\theta ={\displaystyle \frac{\pi }{4}}$ (see [21]).

If $\{e_1,\dots ,e_8\}$ is the canonical basis of $E^8$, we denote $P=e_1$. It is tangent to $M=$ Imx.

A quarter-symmetric connection on $C^4$ can be defined by

$$\overline{\nabla }=\widehat{\overline{\nabla }}+{\lambda }_1\langle Y,e_1\rangle X-{\lambda }_2\langle X,Y\rangle e_1.$$

It follows that all the above inequalities, i.e., Chen first inequality, Chen–Ricci inequality and generalized Euler inequality, are true on M.

These inequalities can be verified by straightforward, but long calculations.

7. Conclusions

Complex manifolds provide a natural and generalized framework for differential geometry, particularly in the study of structures compatible with complex or Hermitian metrics. On the other hand, quarter-symmetric connections offer an alternative geometric setting in which the Levi–Civita connection is supplemented by additional torsion or non-metricity components, leading to richer curvature behavior.

Kählerian slant submanifolds constitute an important class of submanifolds characterized by a special symmetry condition involving the complex structure. In this article, we establish a symmetry condition for the second fundamental form when the submanifold is tangent to P. Furthermore, we investigate intrinsic–extrinsic relations for Kählerian slant submanifolds in complex space forms endowed with a quarter-symmetric connection.

This work can be extended in several directions. One possible extension is to study Kählerian slant submanifolds that are not tangent to P. Another is to develop generalized Euler-type inequalities within this geometric setting.

On the other hand, Calabi–Yau manifolds play a significant role in theoretical physics as they provide an essential geometric framework for string theory. In [31], the authors studied isometric immersions of Lagrangian submanifolds into Calabi–Yau manifolds. Their results suggest further extensions, such as investigating Chen-type inequalities and related problems for submanifolds immersed in Calabi–Yau ambient spaces.

Moreover, other researchers in the field could continue our study and deal with geodesic mappings of equiaffine and Ricci symmetric spaces, geometry in the large of Einstein-like manifolds, and linearly independent curvature tensors of half-symmetric affine connections.