Pinching Results for Submanifolds in Lorentzian–Sasakian Manifolds Endowed with a Semi-Symmetric Non-Metric Connection

Abstract

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form.

1. Introduction

Friedmann and Schouten introduced the concept of semi-symmetric linear and metric connections on differentiable manifolds in [1], while H. A. Hayden independently proposed the same in [2]. K. Yano further dealt with the properties of Riemannian manifolds endowed with a semi-symmetric metric connection in [3]. Agashe and Chafle, in their works [4,5], introduced the notion of a semi-symmetric non-metric connection on a Riemannian manifold and explored submanifolds within this context.

Addressing Chen’s open problem, which involves identifying optimal relationships between intrinsic and extrinsic invariants of a Riemannian submanifold—a key challenge in the geometry of submanifolds—Chen defined the $\delta$-invariants, well known as Chen invariants, in [6,7]. In this respect, many authors have explored Chen’s theory in different ambient spaces, concentrating on specific types of submanifolds. For additional details, we refer to [8,9,10,11,12].

The principal extrinsic invariant is the mean curvature. The Casorati curvature of a submanifold in a Riemannian manifold is also an extrinsic invariant. This curvature is defined as the normalized square of the length of the second fundamental form of the submanifold, extending the concept of the principal directions of a hypersurface in a Riemannian manifold. Therefore, obtaining optimal inequalities for the Casorati curvatures of submanifolds in different manifolds is of significant interest. Decu et al. [13] derived optimal inequalities for the Casorati curvature in terms of the scalar curvature on a submanifold in a Riemannian space form. Several authors have obtained geometric inequalities for the Casorati curvatures of submanifolds, as evidenced by works such as [13,14,15,16].

In particular, discussions on the Chen basic inequality and the inequality of Casorati invariants for submanifolds in different ambient spaces, equipped with a semi-symmetric metric connection or a semi-symmetric non-metric one, can be found in [17,18,19,20,21,22,23,24,25].

In the present paper, we establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, for slant submanifolds in a Lorentzian–Sasakian space form equipped with a semi-symmetric non-metric connection. Also, we prove an improved Chen basic inequality for Legendrian submanifolds in such spaces. Examples of slant submanifolds in a Lorentzian–Sasakian space form are presented.

2. Preliminaries

Let $(\tilde{M},g)$ be a Riemannian manifold, $dim\tilde{M}=n+q$. A linear connection $\tilde{\nabla }$ on $\tilde{M}$ is called a semi-symmetric connection if the torsion tensor field $\tilde{T}$, given by

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

for any vector fields X and Y on $\tilde{M}$, satisfies

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

(1)

where $\omega$ is the 1-form associated with the vector field P on $\tilde{M}$, i.e., $\omega \left(X\right)=g(X,P)$.

If $\tilde{\nabla }g=0$, then $\tilde{\nabla }$ is said to be a semi-symmetric metric connection. Otherwise, i.e., if $\tilde{\nabla }g\ne 0$, $\tilde{\nabla }$ is said to be a semi-symmetric non-metric connection.

Let $\stackrel{\circ }{\tilde{\nabla }}$ be the Levi-Civita connection with respect to the Riemannian metric g. Following [4], a semi-symmetric non-metric connection $\tilde{\nabla }$ on M is given by

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

for any vector fields X and Y tangent to $\tilde{M}$.

In the following, we consider $(\tilde{M},g)$ be a Riemannian manifold endowed with a semi-symmetric non-metric connection $\tilde{\nabla }$ and the Levi-Civita connection $\stackrel{\circ }{\tilde{\nabla }}$, and we consider M to be a submanifold in $\tilde{M}$.

Let $\tilde{R}$ and $\stackrel{\circ }{\tilde{R}}$ be the curvature tensor fields of the Riemannian manifold $\tilde{M}$ corresponding to $\tilde{\nabla }$ and $\stackrel{\circ }{\tilde{\nabla }}$, respectively. Then, $\tilde{R}$ is expressed by (see [4])

$$\tilde{R}(X,Y,Z,W)=\stackrel{\circ }{\tilde{R}}(X,Y,Z,W)+s(X,Z)g(Y,W)-s(Y,Z)g(X,W),$$

(2)

for any vector fields $X,Y,Z$, and W on $\tilde{M}$, where s is $(0,2)$ tensor given by

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

The Gauss formulae for the connections $\tilde{\nabla }$ and $\stackrel{\circ }{\tilde{\nabla }}$ are written as

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

for any vector fields X and Y on the submanifold M, where $\stackrel{\circ }{h}$ is the second fundamental form of M and h is a $(0,2)$-tensor on M. From [5], it is known that $\stackrel{\circ }{h}=h$.

Let $\tilde{M}$ be a differentiable manifold of dimension $(2m+1)$, where $m\ge 1$, admitting a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$, and an l-form $\eta$ such that

$${\varphi }^2\left(X\right)=-X+\eta \left(X\right)\xi ,\varphi \xi =0,\eta \left(\varphi X\right)=0,\eta \left(\xi \right)=1,$$

(3)

for any vector field X on $\tilde{M}$.

We say that $\tilde{M}$ has an almost contact structure $(\varphi ,\xi ,\eta )$ and call $\tilde{M}$ an almost contact manifold.

Since $\tilde{M}$ has a globally defined vector field $\xi$, it is able to admit a Lorentzian metric g such that $g(\xi ,\xi )=-1$.

If M admits a normal almost contact structure $(\varphi ,\xi ,\eta )$ and a Lorentzian metric g with

$$\begin{array}{cc}\hfill g(\varphi X,\varphi Y)& =g(X,Y)+\eta \left(X\right)\eta \left(Y\right),({\stackrel{\circ }{\tilde{\nabla }}}_X\eta )\left(Y\right)=g(\varphi X,Y),\hfill \end{array}$$

where $\stackrel{\circ }{\tilde{\nabla }}$ is the Levi-Civita connection of g, then M is called a Sasakian manifold with a Lorentzian metric or simply a Lorentzian–Sasakian manifold.

On such a manifold, one also has

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

A Lorentzian–Sasakian manifold having a constant $\varphi$-sectional curvature c is called a Lorentzian–Sasakian space form and is denoted by $\tilde{M}\left(c\right)$.

Theorem 1

([26]). The curvature tensor field of a Lorentzian–Sasakian space form $\tilde{M}\left(c\right)$ has the expression

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

(4)

If $\tilde{M}$ is a $(2m+1)$-dimensional Lorentzian–Sasakian space form of constant $\varphi$-sectional curvature c endowed with a semi-symmetric non-metric connection, then, from expressions (2) and (4), it follows that the curvature tensor field $\tilde{R}$ becomes

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

(5)

Denoting the tangent bundle as $TM$ and the normal bundle of M as $T^{\perp }M$, we can write

$$\varphi X=PX+FX,\varphi N=tN+fN,$$

where $PX$ and $FX$ denote the tangential component and the normal component, respectively, of $\varphi X$, and $tN$ and $fN$ are the tangential and the normal components, respectively, of $\varphi N$.

We decompose the vector field P on M into its tangent and normal components $P^{\top }$ and $P^{\perp }$, respectively; one has $P=P^{\top }+P^{\perp }$.

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

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

(6)

for all vector fields X, Y, Z, and W on M.

Recently, one of the authors of the present work and A. Mihai [27] defined the sectional curvature of a non-metric connection. Let $p\in M$, $\pi \subset T_{p}M$ be a two-plane section at p, and $\{e_1,e_2\}$ be an orthonormal basis of $\pi$.

Since $R(X,Y,Z,W)\ne R(X,Y,W,Z)$, we cannot define the sectional curvature by the standard definition.

The sectional curvature $K\left(\pi \right)$, with respect to the induced connection ∇, is defined as follows [27]:

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

(7)

For any orthonormal basis $\{e_1,\cdots ,e_n\}$ of the tangent space $T_{p}M$, the scalar curvature $\tau$ at p with respect to the semi-symmetric non-metric connection 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 $e_i$ and $e_j$.

In particular, if M is a submanifold tangent to $\xi$ of a Lorentzian–Sasakian manifold equipped with a semi-symmetric non-metric connection, for an orthonormal basis $\{e_1,e_2,\cdots ,e_n,e_{n+1}=\xi \}$ of $T_{p}M$, the scalar curvature $\tau$ of M at p, from (8), becomes

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

(9)

The normalized scalar curvature $\rho$ is given by

$$\begin{array}{c}\hfill \rho ={\displaystyle \frac{2\tau }{n(n+1)}}.\end{array}$$

(10)

The mean curvature vector $H\left(p\right)$ at $p\in M$ is given by

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

(11)

We denote

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

(12)

where ${\parallel P\parallel }^2$ and ${\parallel F\parallel }^2$ are independent of the choice of the above orthonormal basis.

Denote by $h_{ij}^r=g(h(e_i,e_j),e_r)$, $i,j=1,...,n+1$, $r\in \{n+2,...,2m+1\}$ the components of the second fundamental form; then, the square norm of h is given by

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

Lemma 1.

Let $\tilde{M}$ be a Lorentzian–Sasakian manifold and M a submanifold tangent to ξ. Then,

(i) 

$h(\xi ,\xi )=0;$

(ii) 

$h(X,\xi )=-FX$, for any vector field X on M.

Proof. 

Let $p\in M$ and $X\in T_{p}M$. Then, ${\stackrel{\circ }{\tilde{\nabla }}}_X\xi =-\varphi X=-PX-FX.$

On the other hand, by the Gauss formula,

$${\stackrel{\circ }{\tilde{\nabla }}}_X\xi ={\stackrel{\circ }{\nabla }}_X\xi +h(X,\xi ).$$

Then, one has $h(X,\xi )=-FX.$

For $X=\xi$, we obtain (i), and for X orthogonal to $\xi$, one obtains (ii). □

In [28], A. Lotta defined the following class of submanifolds. Let $\tilde{M}$ be an almost contact metric manifold. A submanifold M tangent to $\xi$ is called a contact slant submanifold if, for all $p\in M$ and $X\in T_{p}M$ linearly independent of $\xi$, $\varphi X$ makes a constant angle $\theta$ with $T_{p}M$. In this case, $\theta \in [0,\pi /2]$ is called the slant angle of M. As particular cases, we mention the invariant submanifolds ($\theta =0$) and the anti-invariant submanifolds ($\theta =\pi /2$).

A contact slant submanifold is said to be proper if it is neither invariant nor anti-invariant.

A proper contact slant submanifold is said to be a special contact slant submanifold if

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

for all vector fields X and Y on M.

This represents the contact analogue of a Kählerian slant submanifold defined by B.Y. Chen in [29]. Such a manifold is Kählerian with respect to a suitable complex structure.

On a special contact slant submanifold, $A_{FX}Y=A_{FY}X$, for all space-like vector fields X and Y (see [9]).

Definition 1.

A submanifold M is

(a) 

totally geodesic if the second fundamental form vanishes identically;

(b) 

totally umbilical if $h(X,Y)=g(X,Y)H$, for any tangent vectors X and Y on M;

(c) 

called minimal if $H=0$.

A point p in an n-dimensional submanifold M of an m-dimensional Riemannian manifold $\tilde{M}$ is said to be an invariantly quasi-umbilical point if there are $m-n$ mutually orthogonal unit vectors ${\xi }_{n+1},\cdots ,{\xi }_m$ normal to M such that the shape operators with respect to all directions ${\xi }_{\alpha }$ have an eigenvalue of multiplicity $n-1$, and for each ${\xi }_{\gamma }$, the distinguished eigenvalue direction is the same. The submanifold M is an invariantly quasi-umbilical submanifold if all its points are invariantly quasi-umbilical points.

3. Example

Our goal is to find interesting examples of special contact slant submanifolds in Lorentzian–Sasakian space forms.

First, we state the following result:

Theorem 2.

Let S be a proper slant surface of ${\mathbb{C}}^2$, defined by the equation

$$x(u,v)=(f_1(u,v),f_2(u,v),f_3(u,v),f_4(u,v)),$$

(13)

with $\partial /\partial x$ and $\partial /\partial y$ non-null and orthogonal. Then,

$$y(u,v,t)=2(f_1(u,v),f_2(u,v),f_3(u,v),f_4(u,v),t)$$

(14)

defines a 3-dimensional contact slant submanifold M of $({\mathbb{R}}^5,{\varphi }_0,\eta ,\xi ,g)$, such that if we put

$$\begin{array}{cc}\hfill e_1& ={\displaystyle \frac{\partial }{\partial u}}+(2f_3{\displaystyle \frac{\partial f_1}{\partial u}}+2f_4{\displaystyle \frac{\partial f_2}{\partial u}}){\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill e_2& ={\displaystyle \frac{\partial }{\partial v}}+(2f_3{\displaystyle \frac{\partial f_1}{\partial v}}+2f_4{\displaystyle \frac{\partial f_2}{\partial v}}){\displaystyle \frac{\partial }{\partial t}},\hfill \end{array}$$

then $\{e_1,e_2,\xi \}$ is an orthonormal frame on M.

We consider on ${\mathbb{R}}^{2m+1}$ the following Lorentzian–Sasakian structure $({\mathbb{R}}^{2m+1},{\varphi }_0,\eta ,\xi ,g)$, given by

$$\begin{array}{c}\hfill \eta ={\displaystyle \frac{1}{2}}\left(dz-\sum _{i=1}^m{y}^{i}d{x}^i\right),\xi =2{\displaystyle \frac{\partial }{\partial z}},\\ \hfill g=-\eta \otimes \eta +{\displaystyle \frac{1}{4}}\sum _{i=1}^m(d{x}^i\otimes d{x}^i+d{y}^i\otimes d{y}^i),\end{array}$$

$$\begin{array}{cc}\hfill {\varphi }_0\left(\sum _{i=1}^m\left(X_i{\displaystyle \frac{\partial }{\partial x^i}}+Y_i{\displaystyle \frac{\partial }{\partial y^i}}\right)+Z{\displaystyle \frac{\partial }{\partial z}}\right)& =\sum _{i=1}^m\left(Y_i{\displaystyle \frac{\partial }{\partial x^i}}-X_i{\displaystyle \frac{\partial }{\partial y^i}}\right)+\sum _{i=1}^m{Y}_i{y}^i{\displaystyle \frac{\partial }{\partial z}},\hfill \end{array}$$

where $\{x^i,y^i,z\}$, $i=1,\cdots ,m$, are the Cartesian coordinates.

Example 1.

The equation $x(u,v,t)=2(u+v,kcosv,v-u,ksinv,t)$ defines a special contact slant submanifold with slant angle ${cos}^{-1}\sqrt{2+k^2}$.

Proof. 

We have

$$e_1={\displaystyle \frac{1}{\sqrt{2}}}(1,0,-1,0,0),$$

(15)

$$e_2={\displaystyle \frac{1}{\sqrt{k^2+2}}}(1,-ksinv,1,kcosv,0).$$

(16)

We compute the slant angle

$$cos\theta =g({\varphi }_0{e}_2,e_1)=-g(e_2,{\varphi }_0{e}_1)=\sqrt{{\displaystyle \frac{2}{2+k^2}}}$$

Denote $e_i^{*}={\displaystyle \frac{F{e}_i}{\parallel F{e}_i\parallel }}$, $i-1,2$. By Lemma 1, $h(e_1,\xi )=-sin\theta e_2^{*}$ and $h(e_2,\xi )=0$. A straightforward calculation gives $h(e_1,e_2)=-sin\theta e_1^{*}$. Then, the matrix of the second fundamental form is

$$\left[h(e_i,e_j)\right]=\left(\begin{array}{ccc}0& -sin\theta e_1^{*}& -sin\theta e_2^{*}\\ -sin\theta e_1^{*}& {\displaystyle \frac{1}{2k^2+8}}\left[2(0,-kcosv,0,-ksinv,0)\right]& 0\\ -sin\theta e_2^{*}& 0& 0\end{array}\right).$$

□

4. A Chen Inequality for the Scalar Curvature

Our main interest was motivated by certain inequalities obtained by B.Y. Chen.

Regarding the mean curvature H and the scalar curvature $\tau$, B.Y. Chen [30] proved that, on an n-dimensional submanifold M in a Riemannian space form of constant sectional curvature c,

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

The equality holds identically if and only if M is totally umbilical.

In this section, we give an improved Chen inequality for the scalar curvature of a special contact slant submanifold in a Lorentzian–Sasakian space form equipped with a semi-symmetric non-metric connection.

Recall that an $(n+1)$-dimensional submanifold M tangent to $\xi$ of a $(2n+1)$-dimensional Lorentzian–Sasakian manifold $\tilde{M}$ is a contact H-umbilical submanifold if the second fundamental form has the form

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

(17)

for some smooth functions $\alpha$ and $\beta$ with respect to some suitable orthonormal local frame fields $\{e_1,\dots ,e_n,e_{n+1}=\xi \}$ on $TM$ and $\{e_1^{*},\dots ,e_n^{*}\}$ on $T^{\perp }M$.

Theorem 3.

Let $\tilde{M}\left(c\right)$ be a Lorentzian–Sasakian space form of dimension $2n+1$$(n\ge 2)$ equipped with a semi-symmetric non-metric connection $\tilde{\nabla }$ and M an $(n+1)$-dimensional special contact slant submanifold. Then,

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

(18)

where $\lambda ={\sum }_{j=1}^{n}s(e_j,e_j)$.

Moreover, the equality sign holds identically if and only if M is a contact H-umbilical submanifold where $\alpha =3\beta$.

Proof. 

Let M be an $(n+1)$-dimensional special contact slant submanifold of a $(2n+1)$-dimensional Lorentzian–Sasakian space form $\tilde{M}\left(c\right)$ equipped with a semi-symmetric non-metric connection $\tilde{\nabla }$. Let $p\in M$ and $\{e_1,\cdots ,e_n,e_{n+1}=\xi \}$ be an orthonormal basis of $T_{p}M$. An orthonormal basis of $T_p^{\perp }M$ is given by $\{e_1^{*},\cdots ,e_n^{*}\}$, where $e_k^{*}={\displaystyle \frac{1}{sin\theta }}F{e}_k$, $k=1,\cdots ,n$.

We use the formula

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

(19)

If we put $X=W=e_i$ and $Y=Z=e_j$, $i\ne j$, $i,j=1,\cdots ,n$ in the Gauss equation, then

$$\begin{array}{cc}\hfill R(e_i,e_j,e_j,e_i)=& \tilde{R}(e_i,e_j,e_j,e_i)-g(h(e_i,e_j),h(e_i,e_j))\hfill \\ \hfill & +g(h(e_i,e_i),h(e_j,e_j))-g(P^{\perp },h(e_j,e_j)).\hfill \end{array}$$

It follows that the scalar curvature is given by

$$\begin{array}{cc}\hfill 2\tau \left(p\right)=& \sum _{1\le i\ne j\le n}\tilde{R}(e_i,e_j,e_j,e_i)+\sum _{1\le i\ne j\le n}[g(h(e_i,e_i),h(e_j,e_j))-g(h(e_i,e_j),h(e_i,e_j))]\hfill \\ \hfill & -\sum _{1\le i\ne j\le n}g(P^{\perp },h(e_j,e_j))+2\sum _{j=1}^{n}K(\xi \wedge e_j).\hfill \end{array}$$

(20)

We calculate $\tilde{R}(e_i,e_j,e_j,e_i)$ using Formula (5) of the curvature tensor, for $i,j=1,\cdots n,i\ne j$.

$$\begin{array}{cc}\hfill \tilde{R}(e_i,e_j,e_j,e_i)=& {\displaystyle \frac{c-3}{4}}\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\}\hfill \\ \hfill & +{\displaystyle \frac{c+1}{4}}[g(e_i,\varphi e_j)g(\varphi e_j,e_i)-g(e_j,\varphi e_j)g(\varphi e_i,e_i)\hfill \\ \hfill & +2g(e_i,\varphi e_j)g(\varphi e_j,e_i)\hfill \\ \hfill & +\eta \left(e_j\right)\eta \left(e_j\right)g(e_i,e_i)-\eta \left(e_i\right)\eta \left(e_j\right)g(e_j,e_i)\hfill \\ \hfill & +g(e_i,e_j)\eta \left(e_j\right)\eta \left(e_i\right)-g(e_j,e_j)\eta \left(e_i\right)\eta \left(e_i\right)]\hfill \\ \hfill & +s(e_i,e_j)g(e_j,e_i)-s(e_j,e_j)g(e_i,e_i).\hfill \end{array}$$

(21)

A straightforward calculation implies

$$\begin{array}{c}\hfill \tilde{R}(e_i,e_j,e_j,e_i)={\displaystyle \frac{c-3}{4}}+3{\displaystyle \frac{c+1}{4}}{g}^2(\varphi e_i,e_j)-s(e_j,e_j).\end{array}$$

(22)

We denote $\lambda ={\sum }_{i=1}^{n}s(e_i,e_i)$.

Substituting (22) in (20), we obtain

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n-1){\displaystyle \frac{c-3}{4}}+3{\displaystyle \frac{c+1}{4}}\sum _{i\ne j}{g}^2(\varphi e_i,e_j)-(n-1)\lambda \hfill \\ \hfill & -{\parallel h\parallel }^2+{(n+1)}^2{\parallel H\parallel }^2-(n+1)(n-1)\omega \left(H\right)+2\sum _{j=1}^{n}K(\xi \wedge e_j).\hfill \end{array}$$

(23)

From the definition of the sectional curvature and because $\xi$ is time-like, we have

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

(24)

If we put $X=W=\xi$ and $Y=Z=e_j$, for $j=1,\cdots n$ in the Gauss equation, then we obtain

$$\begin{array}{cc}\hfill R(\xi ,e_j,e_j,\xi )& =\tilde{R}(\xi ,e_j,e_j,\xi )\hfill \\ \hfill & +g(h(\xi ,\xi ),h(e_j,e_j))-g(h(\xi ,e_j),h(\xi ,e_j))\hfill \\ \hfill & -g(P^{\perp },h(e_j,e_j))g(\xi ,\xi ).\hfill \end{array}$$

(25)

We calculate $\tilde{R}(\xi ,e_j,e_j,\xi )$ using Formula (5) of the curvature tensor.

$$\begin{array}{c}\hfill \tilde{R}(\xi ,e_j,e_j,\xi )=-1-s(e_j,e_j).\end{array}$$

(26)

By using Equation (26) and Lemma 1, we have $h(\xi ,\xi )=0$ and $h(\xi ,e_j)=-F{e}_j$. Obviously, $\parallel F{e}_j\parallel =sin\theta$. Then, Equation (25) becomes

$$\begin{array}{c}\hfill R(\xi ,e_j,e_j,\xi )=-1-s(e_j,e_j)-{sin}^2\theta +g(P^{\perp },h(e_j,e_j)).\end{array}$$

(27)

Similarly, we have

$$\begin{array}{c}\hfill R(e_j,\xi ,\xi ,e_j)=-1-s(\xi ,\xi )-{sin}^2\theta .\end{array}$$

(28)

We substitute (27) and (28) in (24); taking the summation, we obtain

$$\begin{array}{c}\hfill \sum _{j=1}^{n}K(\xi \wedge e_j)=n(1+{sin}^2\theta )+{\displaystyle \frac{\lambda }{2}}+{\displaystyle \frac{n}{2}}s(\xi ,\xi )+{\displaystyle \frac{1}{2}}(n+1)\omega \left(H\right).\end{array}$$

(29)

If we put (29) in (23), we obtain

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n-1){\displaystyle \frac{c-3}{4}}+3{\displaystyle \frac{c+1}{4}}\sum _{1\le i\ne j\le n}{g}^2(\varphi e_i,e_j)\hfill \\ \hfill & -(n-2)\lambda +ns(\xi ,\xi )-(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta )\hfill \\ \hfill & -{\parallel h\parallel }^2+{(n+1)}^2{\parallel H\parallel }^2.\hfill \end{array}$$

(30)

We denote by $h_{ij}^k=g(h(e_i,e_j),e_k^{*})$, $1\le i,j,k\le n$.

Since M is a special contact slant submanifold in $\tilde{M}\left(c\right)$, we know from [9] that the shape operator satisfies $A_{FX}Y=A_{FY}X$, for any space-like vector fields $X,Y\in \Gamma \left(TM\right)$.

The above relation implies the symmetry of the coefficients $h_{ij}^k$ of the second fundamental form, i.e.,

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

(31)

By the definition of the mean curvature, we obtain

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

(32)

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

From Equations (30)–(32),

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n-1){\displaystyle \frac{c-3}{4}}+3{\displaystyle \frac{c+1}{4}}\sum _{i\ne j}{g}^2(\varphi e_i,e_j)\hfill \\ \hfill & -(n-2)\lambda +ns(\xi ,\xi )-(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta )\hfill \\ \hfill & +2\sum _i^n\sum _{1\le j<k\le n}{h}_{jj}^i{h}_{kk}^i-2\sum _{1\le i\ne j\le n}{\left(h_{jj}^i\right)}^2-6\sum _{1\le i<j<k\le n}{\left(h_{ij}^k\right)}^2.\hfill \end{array}$$

(33)

Let us introduce a parameter, $m={\displaystyle \frac{n+2}{n-1}}$, where $n\ge 2$. For studying the inequality of ${\parallel H\parallel }^2$, we follow the technique used in [31]. Then, we obtain

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

(34)

For contact slant submanifolds, we have ${\parallel P\parallel }^2=n{cos}^2\theta$; then, we obtain

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

(35)

The equality holds if and only if

$h_{ij}^k=0$, for all $1\le i<j<k\le n$;

$h_{ii}^i=3h_{jj}^i$, for all $1\le i\ne j\le n.$

We may choose $e_1,...,e_n$ in such a way that $e_1^{*}$ is parallel to H; then, $h_{kk}^j=0$, for $j>1$, $k=1,...,n$.

It follows that M is a contact H-umbilical submanifold where $\alpha =3\beta$. □

5. Inequalities for the Casorati Curvatures

Let $\tilde{M}$ be an m-dimensional Lorentzian manifold and M an n-dimensional submanifold. The Casorati curvature $\mathcal{C}$ of M is expressed by

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

(36)

For any subspace L in $T_{p}M$ of dimension $r⩾2$ and $\{e_1,e_2,...,e_r\}$ an orthonormal basis of L, its scalar curvature $\tau \left(L\right)$ is

$$\tau \left(L\right)=\sum _{1\le \alpha <\beta \le r}K(e_{\alpha }\wedge e_{\beta }).$$

(37)

The Casorati curvature $\mathcal{C}\left(L\right)$ of L is given by

$$\mathcal{C}\left(L\right)={\displaystyle \frac{1}{r}}\sum _{\gamma =n+1}^m\sum _{i,j=1}^r{\left(h_{ij}^{\gamma }\right)}^2.$$

(38)

For n-dimensional submanifolds, S. Decu, S. Haesen, and L. Verstraelen introduced in [13] the notion of normalized Casorati $\delta$-curvatures ${\delta }_c(n-1)$ and ${\delta }_c^{*}(n-1)$ as follows:

$${\left[{\delta }_c(n-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{n+1}{2n}}inf\left\{\mathcal{C}\left(L\right)\right|L\mathtt{a}\mathtt{hyperplane}\mathtt{of}{T}_{p}M\},$$

$${\left[{\delta }_c^{*}(n-1)\right]}_p=2{\mathcal{C}}_p-{\displaystyle \frac{2n-1}{2n}}sup\left\{\mathcal{C}\left(L\right)\right|L\mathtt{a}\mathtt{hyperplane}\mathtt{of}{T}_{p}M\}.$$

The same authors [13] proved that, on an n-dimensional submanifold in an m-dimensional real space form with constant sectional curvature c,

$$\rho \le {\delta }_c(n-1)+c,(\mathrm{resp}.\rho \le {\delta }_c^{*}(n-1)+c),$$

(39)

where $\rho$ denotes the normalized scalar curvature.

Moreover, the equality in (39) holds identically if and only if there exists an orthonormal frame $e_1,e_2,\cdots ,e_n,e_{n+1},\cdots ,e_m$ such that, with respect to this frame, the shape operator is $A_{n+1}=\left(\begin{array}{cc}\lambda I_{n-1}& 0\\ 0& 2\lambda \end{array}\right),$ and $A_{n+2}=\cdots =A_m=0$.

We obtain inequalities for the Casorati curvatures of contact slant submanifolds in a Lorentzian–Sasakian manifold equipped with a semi-symmetric non-metric connection.

The normalized Casorati curvatures ${\delta }_c\left(n\right)$ and ${\delta }_c^{*}\left(n\right)$ are defined as (see [32])

$${\left[{\delta }_c\left(n\right)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{n+2}{2(n+1)}}inf\left\{\mathcal{C}\left(L\right)\right|L\mathtt{a}\mathtt{hyperplane}\mathtt{of}{T}_{p}M\},$$

$${\left[{\delta }_c^{*}\left(n\right)\right]}_p=2{\mathcal{C}}_p-{\displaystyle \frac{2n+1}{2(n+1)}}sup\left\{\mathcal{C}\left(L\right)\right|L\mathtt{a}\mathtt{hyperplane}\mathtt{of}{T}_{p}M\}.$$

Theorem 4.

Let M be an $(n+1)$-dimensional, $n+1\ge 3$, contact slant submanifold in a $(2n+1)$-dimensional Lorentzian–Sasakian space form $\tilde{M}\left(c\right)$ endowed with a semi-symmetric non-metric connection $\tilde{\nabla }$. Then, the following result:

(i) 

The normalized δ-curvature ${\delta }_c\left(n\right)$ satisfies

$$\begin{array}{cc}\hfill \rho & ⩽{\delta }_c\left(n\right)+{\displaystyle \frac{n-1}{n+1}}{\displaystyle \frac{c-3}{4}}\hfill \\ \hfill & +3{cos}^2\theta {\displaystyle \frac{c+1}{4(n+1)}}-{\displaystyle \frac{n-2}{n(n+1)}}\lambda +{\displaystyle \frac{1}{n+1}}s(\xi ,\xi )\hfill \\ \hfill & -{\displaystyle \frac{n-2}{n}}\omega \left(H\right)+{\displaystyle \frac{2}{n+1}}(1+{sin}^2\theta ).\hfill \end{array}$$

(40)

(ii) 

The normalized δ-curvature ${\delta }_c^{*}\left(n\right)$ satisfies

$$\begin{array}{cc}\hfill \rho & ⩽{\delta }_c^{*}\left(n\right)+{\displaystyle \frac{n-1}{n+1}}{\displaystyle \frac{c-3}{4}}+3{cos}^2\theta {\displaystyle \frac{c+1}{4(n+1)}}-{\displaystyle \frac{n-2}{n(n+1)}}\lambda +{\displaystyle \frac{1}{n+1}}s(\xi ,\xi )\hfill \\ \hfill & -{\displaystyle \frac{n-2}{n}}\omega \left(H\right)+{\displaystyle \frac{2}{n+1}}(1+{sin}^2\theta ).\hfill \end{array}$$

(41)

Moreover, the equality holds in one of the equations, Equation (40) or (41), if and only if M is an invariantly quasi-umbilical submanifold with flat normal connection in $\tilde{M}\left(c\right)$.

Proof. 

Let $\tilde{M}\left(c\right)$ be a $(2n+1)$-dimensional Lorentzian–Sasakian space form equipped with a semi-symmetric non-metric connection and M an $(n+1)$-dimensional submanifold of $\tilde{M}\left(c\right)$. For any $p\in M$, let $\{e_1,\cdots ,e_{n+1}\}$ be an orthonormal basis of $T_{p}M$ and $\{e_{n+2},\cdots ,e_{2n+1}\}$ an orthonormal basis of $T_p^{\perp }M$.

Equation (30) can be rewritten as

$$\begin{array}{cc}\hfill 2\tau \left(p\right)=& {(n+1)}^2{\parallel H\parallel }^2-(n+1)\mathcal{C}+n(n-1){\displaystyle \frac{c-3}{4}}\hfill \\ \hfill +& {3\parallel P\parallel }^2{\displaystyle \frac{c+1}{4}}-(n-2)\lambda +ns(\xi ,\xi )-(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta ),\hfill \end{array}$$

(42)

where ${\parallel P\parallel }^2=n{cos}^2\theta$.

We use the same technique as in the paper [32].

We define the quadratic polynomial $\mathcal{Q}$ in terms of the second fundamental form

$$\begin{array}{cc}\hfill \mathcal{Q}=& {\displaystyle \frac{1}{2}}n(n+1)\mathcal{C}+{\displaystyle \frac{1}{2}}(n+2)\mathcal{C}\left(L\right)-2\tau \left(p\right)+n(n-1){\displaystyle \frac{c-3}{4}}\hfill \\ \hfill & +{3\parallel P\parallel }^2{\displaystyle \frac{c+1}{4}}-(n-2)\lambda +ns(\xi ,\xi )-(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta ),\hfill \end{array}$$

(43)

where L is a hyperplane of $T_{p}M$. Without loss of generality, we may assume that L is spanned by $e_1,\cdots ,e_n$; then,

$$\begin{array}{cc}\hfill \mathcal{Q}& ={\displaystyle \frac{n}{2}}\sum _{r=n+2}^{2n+1}\sum _{i,j=1}^{n+1}{\left(h_{ij}^r\right)}^2+{\displaystyle \frac{n+1}{2n}}\sum _{r=n+2}^{2n+1}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2-2\tau \left(p\right)\hfill \\ \hfill & +n(n-1){\displaystyle \frac{c-3}{4}}+3{\parallel P\parallel }^2{\displaystyle \frac{c+1}{4}}-(n-2)\lambda +ns(\xi ,\xi )\hfill \\ \hfill & -(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta ).\hfill \end{array}$$

(44)

From (42) and (44), we obtain

$$\begin{array}{c}\hfill \mathcal{Q}={\displaystyle \frac{n+2}{2}}\sum _{r=n+2}^{2n+1}\sum _{i,j=1}^{n+1}{\left(h_{ij}^r\right)}^2+{\displaystyle \frac{n+1}{2n}}\sum _{r=n+2}^{2n+1}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2-\sum _{r=n+2}^{2n+1}{(\sum _{i,j=1}^n{h}_{ij}^r)}^2,\end{array}$$

from which we derive that

$$\begin{array}{cc}\hfill \mathcal{Q}=& \sum _{n+2}^{2n+1}\sum _{i=1}^n\left[{\displaystyle \frac{n^2+n+2}{2n}}{\left(h_{ii}^r\right)}^2+(n+2){\left(h_{in}^r\right)}^2\right]\hfill \\ \hfill & +\sum _{n+2}^{2n+1}\left[{\displaystyle \frac{(n+1)(n+2)}{n}}\sum _{i<j=1}^n{\left(h_{ij}^r\right)}^2-2\sum _{i<j=1}^n{h}_{ii}^r{h}_{jj}^r+{\displaystyle \frac{n}{2}}{\left(h_{nn}^r\right)}^2\right].\hfill \end{array}$$

(45)

The critical points

$$h^c=(h_{11}^{n+2},h_{12}^{n+2},\cdots ,h_{n+1n+1}^{n+2},\cdots ,h_{11}^{2n+1},h_{12}^{2n+1},\cdots ,h_{n+1n+1}^{2n+1})$$

(46)

of $\mathcal{Q}$ are the solutions of the following system of linear homogeneous equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \mathcal{Q}}{\partial h_{ii}^r}}={\displaystyle \frac{(n+1)(n+2)}{n}}{h}_{ii}^r-2\sum _{k=1}^n{h}_{kk}^r=0\hfill \\ {\displaystyle \frac{\partial \mathcal{Q}}{\partial h_{n+1n+1}^r}}=n{h}_{n+1n+1}^r-2\sum _{k=1}^n{h}_{kk}^r=0\hfill \\ {\displaystyle \frac{\partial \mathcal{Q}}{\partial h_{ij}^r}}={\displaystyle \frac{2(n+1)(n+2)}{n}}{h}_{ij}^r=0\hfill \\ {\displaystyle \frac{\partial \mathcal{Q}}{\partial h_{in+1}^r}}=2(n+2)h_{in}^r=0\hfill \end{array}\right.$$

(47)

where $i,j\in \{1,2,\cdots ,n\}$, $i\ne j$, and $r\in \{n+2,\cdots ,2n+1\}$.

Therefore, every solution $h^c$ has $h_{ij}^r=0$, for $i\ne j$, and the determinant which corresponds to the first two sets of equations of the above system is zero (there exists a solution for non-totally geodesic submanifolds). Moreover, the Hessian matrix of $H\left(\mathcal{Q}\right)$ has the following eigenvalues:

$$\begin{array}{cc}\hfill & {\lambda }_{11}=0,{\lambda }_{22}={\displaystyle \frac{2n^2-n+2}{n}},\hfill \\ \hfill & {\lambda }_{33}=\cdots ={\lambda }_{n+1n+1}={\displaystyle \frac{(n+1)(n+2)}{n}},\hfill \\ \hfill & {\lambda }_{ij}={\displaystyle \frac{2(n+1)(n+2)}{n}},{\lambda }_{in}=2(n+2),\forall i,j\in \{1,\cdots ,n\},i\ne j.\hfill \end{array}$$

Thus, we know that $\mathcal{Q}$ is parabolic and reaches the minimum $\mathcal{Q}\left(h^c\right)=0$, for each solution $h^c$ of the system (47). It follows that $\mathcal{Q}\ge 0$, which implies that

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& ⩽{\displaystyle \frac{1}{2}}n(n+1)\mathcal{C}+{\displaystyle \frac{1}{2}}(n+2)\mathcal{C}\left(L\right)+n(n-1){\displaystyle \frac{c-3}{4}}\hfill \\ \hfill & +{3\parallel P\parallel }^2{\displaystyle \frac{c+1}{4}}-(n-2)\lambda +ns(\xi ,\xi )\hfill \\ \hfill & -(n+1)(n-2)\omega \left(H\right)+2n(1+{sin}^2\theta ).\hfill \end{array}$$

(48)

We divide all terms by $n(n+1)$. Then,

$$\begin{array}{cc}\hfill \rho ⩽& {\displaystyle \frac{1}{2}}\mathcal{C}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{(n+2)}{n(n+1)}}\mathcal{C}\left(L\right)+{\displaystyle \frac{n-1}{n+1}}{\displaystyle \frac{c-3}{4}}\hfill \\ \hfill & +3{cos}^2\theta {\displaystyle \frac{c+1}{4(n+1)}}-{\displaystyle \frac{n-2}{n(n+1)}}\lambda +{\displaystyle \frac{1}{n+1}}s(\xi ,\xi )\hfill \\ \hfill & -{\displaystyle \frac{n-2}{n}}\omega \left(H\right)+{\displaystyle \frac{2}{n+1}}(1+{sin}^2\theta ),\hfill \end{array}$$

for every hyperplane L tangent to $T_{p}M$.

Taking the infimum over all tangent hyperplanes L, we find

$$\begin{array}{cc}\hfill \rho ⩽& {\delta }_c\left(n\right)+{\displaystyle \frac{n-1}{n+1}}{\displaystyle \frac{c-3}{4}}\hfill \\ \hfill & +3{cos}^2\theta {\displaystyle \frac{c+1}{4(n+1)}}-{\displaystyle \frac{n-2}{n(n+1)}}\lambda +{\displaystyle \frac{1}{n+1}}s(\xi ,\xi )\hfill \\ \hfill & -{\displaystyle \frac{n-2}{n}}\omega \left(H\right)+{\displaystyle \frac{2}{n+1}}(1+{sin}^2\theta ),\hfill \end{array}$$

which is the result.

We remark that the equality sign holds if and only if

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

(49)

$$h_{n+1n+1}=2h_{11}^r=\cdots =2h_{nn}^r,\forall r\in \{n+2,\cdots ,2n+1\}.$$

(50)

From Equations (49) and (50), we conclude that the equality sign holds in the inequality (40) if and only if the submanifold M is invariantly quasi-umbilical with a trivial normal connection in $\tilde{M}\left(c\right)$.

Similarly to the first part, we can establish the inequality for the second part of the theorem. □

6. An Improved Chen First Inequality for Legendrian Submanifolds in a Lorentzian–Sasakian Space Form Endowed with a Semi-Symmetric Non-Metric Connection

An n-dimensional submanifold M normal to $\xi$ in a $(2n+1)$-dimensional Lorentzian– Sasakian manifold is called a Legendrian submanifold.

For Legendrian submanifolds in a Lorentzian–Sasakian space form $\tilde{M}\left(c\right)$ of dimension $2n+1$, admitting a semi-symmetric non-metric connection, we prove a Chen inequality.

Theorem 5.

Let M be an n-dimensional $(n\ge 3)$ Legendrian submanifold isometrically immersed in a Lorentzian–Sasakian space form $\tilde{M}\left(c\right)$, endowed with a semi-symmetric non-metric connection $\tilde{\nabla }$, and $p\in M$, $\pi \subset T_{p}M$ a two-plane section. Then, we have

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)\le & {\displaystyle \frac{n^2(2n-3)}{2(2n+3)}}{\parallel H\parallel }^2+{\displaystyle \frac{(n-2)(n+1)}{2}}{\displaystyle \frac{c-3}{4}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill -{\displaystyle \frac{(n-1)}{2}}\lambda +{\displaystyle \frac{1}{2}}\mathtt{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathtt{trace}\left(h_{|\pi }\right),P).\end{array}$$

(51)

Moreover, the equality case of the inequality holds for some two-plane section π at a point $p\in M$ if and only if there exists an orthonormal basis $\{e_1,\cdots ,e_n\}$ of the tangent space $T_{p}M$ at p such that $\pi =$ 𝚜𝚙𝚊𝚗{e₁, e₂}, and with respect to this basis, the second fundamental form takes the following form:

$$\begin{array}{cc}\hfill h(e_1,e_1)& =a\varphi e_1+3\lambda \varphi e_3,h(e_1,e_3)=3\lambda \varphi e_1,h(e_3,e_j)=4\lambda \varphi e_j.\hfill \\ \hfill h(e_2,e_2)& =-a\varphi e_1+3\lambda \varphi e_3,h(e_2,e_3)=3\lambda \varphi e_2,h(e_j,e_k)=4\lambda \varphi e_3{\delta }_{jk}.\hfill \\ \hfill h(e_1,e_2)& =a\varphi e_2,h(e_3,e_3)=12\lambda \varphi e_3,h(e_1,e_k)=h(e_2,e_j)=0.\hfill \end{array}$$

for some numbers $a,\lambda$, where $j,k=4,\cdots ,n$.

Proof. 

Let $p\in M$ and $\pi \subset T_{p}M$ be a two-plane section and $\{e_1,\cdots ,e_n\}$ be an orthonormal basis of the tangent space $T_{p}M$ at p such that $e_1,e_2\in \pi$. Since M is a Legendrian submanifold, we can choose an orthonormal basis $\{e_1^{*}=e_{n+1}=\varphi e_1,e_2^{*}=e_{n+2}=\varphi e_2,\cdots ,e_n^{*}=e_{2n}=\varphi e_n,e_{2n+1}^{*}=\xi \}$.

For the calculation of $\tilde{R}(e_i,e_j,e_j,e_i)$, we use Formula (5), where we put $X=W=e_i$ and $Y=Z=e_j$. $i,j=1,\cdots n$, $i\ne j$.

$$\begin{array}{cc}\hfill \tilde{R}(e_i,e_j,e_j,e_i)& ={\displaystyle \frac{c-3}{4}}\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\}\hfill \\ \hfill & +{\displaystyle \frac{c+1}{4}}[g(e_i,\varphi e_j)g(\varphi e_j,e_i)-g(e_j,\varphi e_j)g(\varphi e_i,e_i)\hfill \\ \hfill & +2g(e_i,\varphi e_j)g(\varphi e_j,e_i)\hfill \\ \hfill & +\eta \left(e_j\right)\eta \left(e_j\right)g(e_i,e_i)-\eta \left(e_i\right)\eta \left(e_j\right)g(e_j,e_i)\hfill \\ \hfill & +g(e_i,e_j)\eta \left(e_j\right)\eta \left(e_i\right)-g(e_j,e_j)\eta \left(e_i\right)\eta \left(e_i\right)]\hfill \\ \hfill & +s(e_i,e_j)g(e_j,e_i)-s(e_j,e_j)g(e_i,e_i),\hfill \end{array}$$

(52)

which implies that

$$\begin{array}{c}\hfill \tilde{R}(e_i,e_j,e_j,e_i)={\displaystyle \frac{c-3}{4}}-s(e_j,e_j).\end{array}$$

(53)

If we put $X=W=e_i$ and $Y=Z=e_j$ in the Gauss equation for the semi-symmetric non-metric connection, then

$$\begin{array}{cc}\hfill \tilde{R}(e_i,e_j,e_j,e_i)& =R(e_i,e_j,e_j,e_i)+g(h(e_i,e_j),h(e_j,e_i))\hfill \\ \hfill & -g(h(e_i,e_i),h(e_j,e_j))+g(P^{\perp },h(e_j,e_j))g(e_i,e_i)\hfill \\ \hfill & -g(P^{\perp },h(e_i,e_j))g(e_j,e_i).\hfill \end{array}$$

(54)

Then, by substituting (53) in (54), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{c-3}{4}}-s(e_j,e_j)& =R(e_i,e_j,e_j,e_i)+g(h(e_i,e_j),h(e_j,e_i))\hfill \\ \hfill & -g(h(e_i,e_i),h(e_j,e_j))+g(P^{\perp },h(e_j,e_j))g(e_i,e_i)\hfill \\ \hfill & -g(P^{\perp },h(e_i,e_j))g(e_j,e_i),\hfill \end{array}$$

(55)

By summation over $1\le i\ne j\le n$, from the above equation, we have

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

(56)

Since ${\stackrel{\circ }{\nabla }}_X\xi =-\varphi X$, for any $X\in \mathrm{\Gamma }\left(TM\right)$, we have $A_{\xi }X=0$, which implies that $g\left(h\right(X,Y),\xi )=0$.

We denote by $h_{ij}^r=g(h(e_i,e_j),e_r^{*})$, $i,j,r=1,...,n$.

It follows that

$$\begin{array}{cc}\hfill \tau \left(p\right)& =\sum _{r=1}^n\sum _{1\le i<j\le n}\left[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2\right]+n(n-1){\displaystyle \frac{c-3}{8}}\hfill \\ & -{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)-{\displaystyle \frac{(n-1)}{2}}\lambda .\hfill \end{array}$$

(57)

Let $\pi =sp\{e_1,e_2\}$ and by the Gauss formula, if we put $X=W=e_1$ and $Y=Z=e_2$, then

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

(58)

Similarly,

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

(59)

So, from Equations (7), (58), and (59), we have

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

(60)

Equations (57) and (60) imply

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2-h_{11}^r{h}_{22}^r+{\left(h_{12}^r\right)}^2]\hfill \\ \hfill & +n(n-1){\displaystyle \frac{c-3}{8}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)-{\displaystyle \frac{(n-1)}{2}}\lambda \hfill \\ & -{\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{1}{2}}\mathtt{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathtt{trace}\left(h_{|\pi }\right),P).\hfill \end{array}$$

Then, we obtain

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)=& \sum _{r=1}^n\left[\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r-\sum _{j=3}^n[{\left(h_{1j}^r\right)}^2+{\left(h_{2j}^r\right)}^2]-\sum _{3\le i<j\le n}{\left(h_{ij}^r\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{(n-2)(n+1)}{2}}{\displaystyle \frac{c-3}{4}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)-{\displaystyle \frac{n-1}{2}}\lambda \hfill \\ & +{\displaystyle \frac{1}{2}}\mathtt{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathtt{trace}\left(h_{|\pi }\right),P).\hfill \end{array}$$

(61)

Because M is Legendrian, we have $h_{1j}^1=h_{11}^j$, $h_{1j}^j=h_{jj}^1$, for $3\le i\le n$, and $h_{ij}^j=h_{jj}^i$, for $2\le i\ne j\le n$. Then, Equation (61) becomes

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)=& \sum _{r=1}^n\left[\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r-\sum _{j=3}^n[{\left(h_{11}^j\right)}^2+{\left(h_{jj}^1\right)}^2]-\sum _{3\le i<j\le n}{\left(h_{jj}^i\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{(n-2)(n+1)}{2}}{\displaystyle \frac{c-3}{4}}-{\displaystyle \frac{n(n-1)}{2}}\omega \left(H\right)-{\displaystyle \frac{(n-1)}{2}}\lambda \hfill \\ & +{\displaystyle \frac{1}{2}}\mathtt{trace}\left(s_{|\pi }\right)+{\displaystyle \frac{1}{2}}g(\mathtt{trace}\left(h_{|\pi }\right),P).\hfill \end{array}$$

(62)

We will use some ideas from [33] to achieve the proof.

We point out the following inequalities,

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

(63)

for $r=1,2$.

The first inequality in (63) is equivalent to

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

The equality holds if and only if $3h_{jj}^r=h_{11}^r+h_{22}^r,\forall j=3,\cdots ,n$.

The equality holds in the second inequality if and only if $h_{11}^r+h_{22}^r=0$ and $h_{jj}^r=0$,$\forall j=3,\cdots ,n$, and $r=1,2$.

For $r=3,\cdots ,n$, we use the inequality

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

(64)

which is equivalent to

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

The equality holds if and only if

$$\left\{\begin{array}{c}{h}_{11}^r=h_{22}^r=3{\lambda }^r\hfill \\ h_{jj}^r=4{\lambda }^r,\forall j=3,\cdots ,n,j\ne r,r=3,\cdots ,n.\hfill \\ h_{rr}^r=12{\lambda }^r,{\lambda }^r\in \mathbb{R},\hfill \end{array}\right.$$

(65)

From Equations (62)–(64), we obtain the desired inequality (51). □

7. Conclusions

The study of non-metric connections, particularly semi-symmetric non-metric connections, holds considerable importance in differential geometry. For such submanifolds, the sectional curvature cannot be defined in the standard way, because the tensor field $g\left(R\right(X,Y)Z,W)$ is not skew-symmetric in the last two variables.

Recently, in [27], the authors defined a sectional curvature on Riemannian manifolds admitting a semi-symmetric non-metric connection.

In the present paper, we extended this definition in Lorentzian settings, more precisely for Lorentzian-Sasakian manifolds endowed with a semi-symmetric non-metric connection.

Using this sectional curvature, we derived the Chen inequality involving scalar curvature and mean curvature, as well as geometric inequalities for Casorati curvatures, for contact slant submanifolds (which are tangent to the time-like vector $\xi$) in Lorentzian–Sasakian space forms admitting semi-symmetric non-metric connections. In the last section, we established an improved Chen first inequality for Legendrian submanifolds (which are normal to $\xi$) in such space forms.

Future research could extend this work by obtaining additional Chen inequalities or refining the current results for special classes of submanifolds in Lorentzian–Sasakian manifolds or, more generally, in pseudo-Riemannian contact metric manifolds admitting semi-symmetric non-metric connections.