Optimal Inequalities for Submanifolds in Trans-Sasakian Manifolds Endowed with a Semi-Symmetric Metric Connection

Abstract

In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in ($\alpha ,\beta$) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.

1. Introduction

The concept of a semi-symmetric linear connection on a differentiable manifold was defined by A. Friedmann and J.A. Schouten [1]. Afterwards, H.A. Hayden [2] introduced the notion of a semi-symmetric metric connection on a Riemannian manifold. The properties of a Riemannian manifold admitting a semi-symmetric metric connection were investigated by K. Yano [3]. Z. Nakao studied submanifolds of a Riemannian manifold endowed with a semi-symmetric metric connection in [4].

On the other hand, one of the basic problems in the geometry of submanifolds is to find optimal relationships between the extrinsic and intrinsic invariants of a submanifold. In this respect, B.-Y. Chen [5,6] established geometric inequalities, which are known as Chen inequalities. Such inequalities for special classes of submanifolds in different ambient spaces were obtained in [7,8,9,10,11,12,13,14,15,16]. In particular, Chen inequalities for submanifolds of a Riemannian space form admitting a semi-symmetric metric connection were considered in [17,18,19,20].

2. Preliminaries

Let $(\overline{M},g)$ be an m-dimensional Riemannian manifold. A linear connection $\tilde{\nabla }$ on $\overline{M}$ is said to be a semi-symmetric connection if its torsion tensor $\tilde{T}$, defined by

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

(1)

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

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

(2)

where $\omega$ is a 1-form dual to a vector field V on $\overline{M}$, i.e., $\omega \left(\overline{X}\right)=g(\overline{X},V).$

A semi-symmetric connection $\tilde{\nabla }$ is called a semi-symmetric metric connection on $\overline{M}$ if the Riemannian metric g is parallel with respect to $\tilde{\nabla }$, i.e., $\tilde{\nabla }g=0$. It is related to the Levi-Civita connection $\stackrel{\circ }{\tilde{\nabla }}$ on $\overline{M}$ by

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

for any vector fields $\overline{X}$, $\overline{Y}$, on $\overline{M}$, where the vector field P is defined by $g(P,\overline{X})=\phi \left(\overline{X}\right)$, for all vector fields $\overline{X}$.

In the following, let $(\overline{M},g)$ be a a Riemannian manifold endowed with a semi-symmetric metric connection $\tilde{\nabla }$ and the Levi-Civita connection $\stackrel{\circ }{\tilde{\nabla }}$, and M an n-dimensional submanifold of $\overline{M}$. On the submanifold M, one has the Levi-Civita connection $\stackrel{\circ }{\nabla }$ and the induced semi-symmetric metric connection ∇. We denote by $\overline{R}$ the curvature tensor of $\overline{M}$ with respect to $\tilde{\nabla }$ and $\stackrel{\circ }{\tilde{R}}$ the Riemannian curvature tensor of $\overline{M}$ with respect to $\stackrel{\circ }{\tilde{\nabla }}$. The curvature tensors on M of ∇ and $\stackrel{\circ }{\nabla }$, respectively, are denoted by R and $\stackrel{\circ }{R}$.

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

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

for all vector fields $X,Y$ on M, where $h,\stackrel{\circ }{h}$ are the second fundamental forms of M in $\overline{M}$.

The Gauss equation for the Riemannian submanifold M in $\overline{M}$ is

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

(3)

The curvature tensor $\overline{R}$ with respect to the semi-symmetric metric connection $\tilde{\nabla }$ is given by

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)& =\stackrel{\circ }{\overline{R}}(X,Y,Z,W)-\sigma (Y,Z)g(X,W)+\sigma (X,Z)g(Y,W)\hfill \\ \hfill & -\sigma (X,W)g(Y,Z)+\sigma (Y,W)g(X,Z),\forall X,Y,Z,W\in \mathrm{\Gamma }\left(TM\right),\hfill \end{array}$$

(4)

where $\sigma (X,Z)$ is a $(0,2)$-tensor field defined by

$$\sigma (X,Y)=\left({\stackrel{\circ }{\tilde{\nabla }}}_X\omega \right)Y-\omega \left(X\right)\omega \left(Y\right)+\frac{1}{2}\omega \left(P\right)g(X,Y),\forall X,Y\in \mathrm{\Gamma }\left(TM\right)$$

(5)

(see [17]).

Next, we recall some notions and results about almost contact Riemannian manifolds and their submanifolds. For more details, see [21].

A Riemannian manifold $(\overline{M},g)$ of an odd dimension is called an almost contact metric manifold if it admits a $(1,1)$-tensor field $\varphi$, a unit vector field $\xi$, and a 1-form $\eta$ such that

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

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

In addition, on an almost contact metric manifold, we also have

$$\begin{array}{c}\hfill \varphi \xi =0,\eta \left(\varphi X\right)=0,\eta \left(X\right)=g(X,\xi ).\end{array}$$

An almost contact metric manifold is said to be a trans-Sasakian manifold if there are two differentiable functions $\alpha$ and $\beta$ such that

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_X\varphi \right)Y=\alpha (g(X,Y)\xi -\eta \left(Y\right)X)+\beta (g(\varphi X,Y)\xi -\eta \left(Y\right)\varphi X),\end{array}$$

which implies

$${\overline{\nabla }}_X\xi =-\alpha \varphi X+\beta (X-\eta \left(X\right)\xi ).$$

(6)

If $\alpha =1,\beta =0$ a trans-Sasakian manifold is called Sasakian; if $\alpha =0,\beta =1$ a trans-Sasakian manifold is called Kenmotsu; and if $\alpha =0,\beta =0$, a trans-Sasakian manifold is called cosymplectic.

In [22], P. Alegre, D. Blair, and A. Carriazo introduced the notion of a generalized Sasakian space form. It is an almost contact metric manifold $(\overline{M},\varphi ,\xi ,\eta ,g)$ with the curvature tensor satisfying

$$\begin{array}{cc}\hfill \stackrel{\circ }{\overline{R}}(X,Y)Z& =f_1{R}_1(X,Y)Z+f_2{R}_2(X,Y)Z+f_3{R}_3(X,Y)Z,\hfill \end{array}$$

(7)

where

$$\begin{array}{cc}\hfill R_1(X,Y)Z& =g(Y,Z)X-g(X,Z)Y,\hfill \\ \hfill R_2(X,Y)Z& =g(X,\varphi Z)\varphi Yg(Y,\varphi Z)\varphi X+2g(X,\varphi Y)\varphi Z,\hfill \\ \hfill R_3(X,Y)Z& =\eta \left(X\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)X+g(X,Z)\eta \left(Y\right)\xi -g(Y,Z)\eta \left(X\right)\xi ,\hfill \end{array}$$

for any vector fields $X,Y,Z$, with $f_1,f_2,f_3$ differentiable functions on $\overline{M}$. We denote such a manifold by $\overline{M}(f_1,f_2,f_3)$. In particular, we have

(i)

A Sasakian space form for $f_1=\frac{c+3}{4}$ and $f_2=f_3=\frac{c-1}{4}$;

(ii)

A Kenmotsu space form for $f_1=\frac{c-3}{4}$ and $f_2=f_3=\frac{c+1}{4}$;

(iii)

A cosymplectic space form for $f_1=f_2=f_3=\frac{c}{4}$.

If $M(f_1,f_2,f_3)$ is a $(2m+1)$-dimensional generalized Sasakian space form with a semi-symmetric metric connection, then, from (4) and (7), the curvature tensor $\overline{R}$ is given by

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)=& f_1{R}_1(X,Y,Z,W)+f_2{R}_2(X,Y,Z,W)+f_3{R}_3(X,Y,Z,W)\hfill \\ \hfill & -\sigma (Y,Z)g(X,W)+\sigma (X,Z)g(Y,W)\hfill \\ \hfill & -\sigma (X,W)g(Y,Z)+\sigma (Y,W)g(X,Z).\hfill \end{array}$$

(8)

For a submanifold M tangent to $\xi$ of an almost contact metric manifold $\overline{M}$, we write:

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

(9)

for any tangent vector field X and normal vector field N, where $PX,tN$ are the tangent components and $FX,fN$ are the normal components of $\varphi X$ and $\varphi N$.

If $P=0$, then the submanifold is anti-invariant, and if $F=0$, then the submanifold is invariant. The squared norm of P at $p\in M$ is defined as

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

where $e_1,\cdots ,e_n$ is any orthonormal basis of the tangent space $T_{p}M$.

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

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

(10)

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

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

Let $p\in M$ and $\pi \subset T_{p}M$ a 2-plane section. We denote by $K\left(\pi \right)$ the sectional curvature of M with respect to the induced semi-symmetric metric connection $\stackrel{\circ }{\tilde{\nabla }}$.

Let $\{e_1,\cdots ,e_n\}$ be an orthonormal basis of the tangent space $T_{p}M$; then, the scalar curvature $\tau$ at p is defined by

$$\tau \left(p\right)=\sum _{1\le i<j\le n}{k}_{ij},$$

(11)

where $k_{ij}$ is the sectional curvature of the plane section spanned by $e_i$ and $e_j$.

In particular, for an orthonormal basis $\{e_0=\xi ,e_1,e_2,\cdots ,e_{n-1}\}$ of the tangent space $T_{p}M$ at $p\in M$, the scalar curvature $\tau$ at p takes the following form:

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

(12)

The Chen first invariant is defined by

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

(13)

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

In [23], A. Lotta has introduced the notion of contact slant submanifolds into almost contact metric manifolds. A submanifold M tangent to $\xi$ in an almost contact metric manifold $\overline{M}$ is called a contact slant submanifold if for any $p\in M$ and any $X\in T_{p}M$ linearly independent of $\xi$, the angle between $\varphi X$ and $T_{p}M$ is a constant $\theta \in [0,\pi /2]$, called the slant angle of M.

Invariant and anti-invariant submanifolds of $\overline{M}$ are slant submanifolds with slant angle $\theta =0$ and $\theta =\pi /2$, respectively.

A slant immersion which is neither invariant nor anti-invariant is called proper slant.

Definition 1

([24]). A proper contact θ-slant submanifold in a Sasakian manifold $\overline{M}$ is said to be a special contact θ slant submanifold if

$$\left({\nabla }_{X}P\right)Y={cos}^2\theta [g(X,Y)\xi -\eta \left(Y\right)X],\forall X,Y\in \mathrm{\Gamma }\left(TM\right).$$

(14)

Proposition 1

([24]). A submanifold M is a special contact slant submanifold in a Sasakian manifold if and only if for any vector fields $X,Y$ tangent to M, one has

$$A_{FY}X=A_{FX}Y+{sin}^2\theta \tilde{R}(X,Y)\xi .$$

(15)

Corollary 1

([24]). Let M be a special contact slant submanifold of a Sasakian manifold $\overline{M}$. For any $X,Y\in T_{p}M$, $\tilde{R}(X,Y)\xi =0$ if and only if $A_{FY}X=A_{FX}Y$, or equivalently, the coefficients of the second fundamental form satisfy

$$h_{ij}^k=h_{ik}^j=h_{jk}^i,h_{ij}^r=g(h(e_i,e_j),\varphi e_k),$$

(16)

for all $i,j,k$.

It is known that any 3-dimensional proper contact slant submanifold of a Sasakian space form is a special contact slant submanifold [25].

By analogy with the definition of special contact slant submanifolds of a Sasakian manifold and of a Kenmotsu manifold [26], we derive the following definition.

A proper contact slant submanifold M of a trans-Sasakian manifold $\overline{M}$ is called a special contact slant submanifold if

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

for all $X,Y\in \mathrm{\Gamma }\left(TM\right).$

Lemma 1.

Let M be a special contact slant submanifold of a trans-Sasakian manifold $\overline{M}$. Then,

$$A_{FY}X=-{sin}^2\theta \{\alpha [g(X,Y)\xi -\eta \left(Y\right)X]+\beta [g(PX,Y)\xi -\eta \left(Y\right)PX]\}-th(X,Y),$$

for any vector fields $X,Y\in \mathrm{\Gamma }\left(TM\right)$.

It follows that for any vector fields $X,Y,Z$ orthogonal to $\xi$, we have

$$g(A_{FY}Z,X)=g(A_{FY}X,Z)=-g(th(X,Y),Z)=g(h(X,Y),FZ)=g(A_{FZ}Y,X).$$

Let M be an $(n+1)$-dimensional $(n=2k)$ special contact slant submanifold of a $(2n+1)$-dimensional trans-Sasakian manifold $\overline{M}$. An adapted slant frame is given by

$$e_0=\xi ,e_1,e_2=\frac{1}{cos\theta }P{e}_1,....,e_{2k-1},e_{2k}=\frac{1}{cos\theta }P{e}_{2k-1},e_{n+j}=\frac{1}{sin\theta }F{e}_j,$$

for any $j=1,\dots ,n.$

By using Lemma 1, we obtain

Lemma 2.

Let M be an $(n+1)$-dimensional special contact slant submanifold of a $(2n+1)$-dimensional trans-Sasakian manifold $\overline{M}$. Then, with respect to an adapted slant frame, we have

$$\begin{array}{cc}\hfill h_{jk}^i& =h_{ik}^j=h_{ij}^k,i,j,k=1,\dots ,n,\hfill \\ \hfill h_{00}^i& =0,h_{0j}^i=h_{0i}^j=-\alpha sin\theta {\delta }_{ij}.\hfill \end{array}$$

We recall the following:

Lemma 3

([27]). Let M be a contact slant submanifold of a trans-Sasakian generalized Sasakian space form ${\overline{M}}^{2n+1}(f_1,f_2,f_3)$. One can choose

$$e_{1^{*}}=\frac{1}{\parallel H\parallel }H,e_1=-csc\theta t{e}_{1^{*}}.$$

(17)

Definition 2

([28]). A submanifold M of a Sasakian manifold $\overline{M}$ is called a C-totally real submanifold if it is normal to the structure vector field ξ.

Proposition 2.

If M is a C-totally real submanifold of a Sasakian manifold $\overline{M}$, then M is anti-invariant, i.e., $\varphi \left(T_{p}M\right)\subset T_p^{\perp }M,\forall p\in M.$

In particular, if $dim\overline{M}=2dimM+1$, i.e, the dimension of M is maximum, then M is said to be a Legendrian submanifold.

In this case, we may choose an orthonormal basis of $T_p^{\perp }M$ in the following way: $\{e_{n+1}=e_1^{*}=\varphi e_1,\cdots ,e_{2n}=e_n^{*}=\varphi e_n,e_{2m+1}=\xi \}$. Then we have

$$A_{\varphi X}Y=A_{\varphi Y}X,\forall X,Y\in T_{p}M,$$

or equivalently,

$$h_{ij}^k=h_{ik}^j=h_{jk}^i,i,j,k=1,2,\cdots ,n.$$

3. Chen First Inequality for Special Contact Slant Submanifolds

The Chen first inequality for slant submanifolds in Sasakian space forms was established by A. Carriazo [29]. I. Presură [30] improved the above result and obtained the following Chen first inequality for special contact slant submanifolds in Sasakian space forms.

Theorem 1

([30]). Let M be an $(n+1)$-dimensional special contact slant submanifold $(n\ge 2)$ isometrically immersed into a $(2n+1)$-dimensional Sasakian space form $\overline{M}$. Then, for any $p\in M$ and $\pi \subset T_{p}M$ a 2-plane section orthogonal to ξ, we have:

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

In this section, we generalize the inequality from Theorem 1 for special contact slant submanifolds in a $(\alpha ,\beta )$ trans-Sasakian generalized Sasakian space form endowed with a semi-symmetric metric connection.

Theorem 2.

Let M be an $(n+1)$-dimensional $(n\ge 2)$ special contact slant submanifold isometrically immersed in a $(\alpha ,\beta )$ trans-Sasakian generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ of dimension $2n+1$, endowed with a semi-symmetric metric connection with P tangent to M. Then, for any point $p\in M$ and any 2-plane section $\pi \subset T_{p}M$ orthogonal to ξ, we have:

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le \frac{n^2(2n-3)}{2(2n-3)}+\frac{(n+2)(n-1)}{2}{f}_1\hfill \\ \hfill & +\frac{3{cos}^2\theta -2{\psi }^2\left(\pi \right)}{2}{f}_2\hfill \\ \hfill & -n(f_3+{\alpha }^2{sin}^2\theta )-2(n-1)\lambda \hfill \\ \hfill & -n\sigma (e_0,e_0)-\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)],\hfill \end{array}$$

(18)

where ${\psi }^2\left(\pi \right)=g^2(\varphi e_1,e_2)$ and $\lambda ={\sum }_{i=1}^n\sigma (e_i,e_i)=trace\sigma$.

The equality case of the inequality (18) holds at a point $p\in M$ for a plane section π if and only if there exists an orthonormal basis $\{e_0=\xi ,e_1,e_2,\cdots ,e_n\}$ of $T_{p}M$ such that $e_1,e_2\in \pi$ and the second fundamental form takes the following form with respect to this basis.

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

for some real numbers $a,b$ and $j,k=4,\cdots ,n.$

Proof. 

Let M be an $(n+1)$-dimensional special contact slant submanifold of a $(2n+1)$-dimensional trans-Sasakian generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ endowed with a semi-symmetric metric connection. Let $p\in M$ and $\pi \subset T_{p}M$ a 2-plane section orthogonal to $\xi$ with $e_1,e_2\in \pi$ orthonormal vectors and $\{e_0=\xi ,e_1,e_2,\cdots ,e_n\}$ an orthonormal basis of the tangent space $T_{p}M$. An orthonormal basis $\{e_{n+1},e_{n+2},\cdots ,e_{2n}\}$ of the normal space $T_p^{\perp }M$ is given by $F{e}_j=sin\theta e_{n+j}$, $\forall j=1,\cdots ,n.$

In the Gauss equation we put $X=W=e_i$ and $Y=Z=e_j$, $i,j=1,\cdots ,n$; then, the scalar curvature is given by

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

(19)

We calculate $\overline{R}(e_i,e_j,e_j,e_i)$ using the Formula (8) for the curvature tensor and put $X=W=e_i$ and $Y=Z=e_j$, for $i,j=1,\cdots n,i\ne j$.

$$\begin{array}{cc}\hfill \overline{R}(e_i,e_j,e_j,e_i)& =f_1\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\}\hfill \\ \hfill & +f_2\{g(e_i,\varphi e_j)g(\varphi e_j,e_i)-g(e_j,\varphi e_j)g(\varphi e_i,e_i)+2g(e_i,\varphi e_j)g(\varphi e_j,e_i)\}\hfill \\ \hfill & +f_3\{\eta \left(e_i\right)\eta \left(e_j\right)g(e_j,e_i)-\eta \left(e_j\right)\eta \left(e_j\right)g(e_i,e_i)\hfill \\ \hfill & +g(e_i,e_j)\eta \left(e_j\right)\eta \left(e_i\right)-g(e_j,e_j)\eta \left(e_i\right)\eta \left(e_i\right)\}\hfill \\ \hfill & -\sigma (e_j,e_j)g(e_i,e_i)+\sigma (e_i,e_j)g(e_j,e_i)\hfill \\ \hfill & -\sigma (e_i,e_i)g(e_j,e_j)+\sigma (e_j,e_i)g(e_i,e_j),\hfill \end{array}$$

(20)

which implies

$$\begin{array}{c}\hfill \overline{R}(e_i,e_j,e_j,e_i)=f_1+3f_2{g}^2(\varphi e_i,e_j)-\sigma (e_j,e_j)-\sigma (e_i,e_i).\end{array}$$

(21)

By substituting (21) in Equation (19), we find

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n-1)f_1+3f_2\sum _{i\ne j}{g}^2(\varphi e_i,e_j)-2(n-1)\lambda \hfill \\ \hfill & +2\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]+2\sum _{j=1}^{n}K(\xi \wedge e_j),\hfill \end{array}$$

(22)

where $\lambda$ is the trace of $\sigma$.

From the Gauss equation we also have

$$\begin{array}{cc}\hfill & \sum _{j=1}^{n}K(\xi \wedge e_j)=\sum _{j=1}^{n}R(e_0,e_j,e_j,e_0)=\sum _{j=1}^n\overline{R}(e_0,e_j,e_j,e_0)\hfill \\ \hfill & +\sum _{j=1}^{n}g(h(e_0,e_0),h(e_j,e_j))-\sum _{j=1}^{n}g(h(e_0,e_j),h(e_0,e_j)),\hfill \end{array}$$

(23)

or equivalently,

$$\begin{array}{cc}\hfill \sum _{j=1}^{n}K(\xi \wedge e_j)& =\sum _{j=1}^n\overline{R}(e_0,e_j,e_j,e_0)+\sum _{r=1}^n\sum _{j=1}^n[h_{jj}^r{h}_{00}^r-{\left(h_{j0}\right)}^2].\hfill \end{array}$$

(24)

From the Equation (8), it follows that

$$\begin{array}{c}\hfill \sum _{j=1}^n\overline{R}(e_0,e_j,e_j,e_0)=n{f}_1-n{f}_3-n\lambda -n\sigma (e_0,e_0).\end{array}$$

(25)

By using the Lemma 2

$$\begin{array}{c}\hfill \sum _{j=1}^n\sum _{r=1}^n{\left(h_{j0}^r\right)}^2=\sum _{j=1}^n{\left(h_{j\xi }^j\right)}^2=\sum _{r=1}^n{(-\alpha sin\theta )}^2=n{\alpha }^2{sin}^2\theta ,h_{00}^r=0.\end{array}$$

Then, the Equation (24) becomes

$$\begin{array}{cc}\hfill \sum _{j=1}^{n}K(\xi \wedge e_j)& =\sum _{j=1}^{n}R(e_0,e_j,e_j,e_0)=n{f}_1-n{f}_3-n{\alpha }^2{sin}^2\theta -n\lambda -n\sigma (e_0,e_0).\hfill \end{array}$$

(26)

By substituting (26) in (22), we find

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n-1)f_1+3f_2\sum _{i\ne j}{g}^2(\varphi e_i,e_j)-2(n-1)\lambda \hfill \\ \hfill & +2\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & +2n{f}_1-2n{f}_3-2n{\alpha }^2{sin}^2\theta \hfill \\ \hfill & -2n\lambda -2n\sigma (e_0,e_0),\hfill \end{array}$$

(27)

which implies

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =n(n+1)f_1+3f_2\sum _{i\ne j}{g}^2(\varphi e_i,e_j)-2(2n-1)\lambda \hfill \\ \hfill & -2n{f}_3-2n{\alpha }^2{sin}^2\theta -2n\sigma (e_0,e_0)\hfill \\ \hfill & +2\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2].\hfill \end{array}$$

(28)

Let $\pi$ be a 2-plane section of $T_{p}M$ at p, $\pi =sp\{e_1,e_2\}$. From the Gauss equation

$$\begin{array}{cc}\hfill K\left(\pi \right)=R(e_1,e_2,e_2,e_1)=& g(h(e_1,e_1),h(e_2,e_2))-g(h(e_1,e_2),h(e_1,e_2))\hfill \\ \hfill & +\overline{R}(e_1,e_2,e_2,e_1),\hfill \end{array}$$

(29)

or similarly

$$\begin{array}{c}\hfill K\left(\pi \right)=R(e_1,e_2,e_2,e_1)=\sum _{r=1}^n[h_{11}^r{h}_{22}^r-\left(h_{12}^r\right)]+\overline{R}(e_1,e_2,e_2,e_1).\end{array}$$

(30)

Using (8), we obtain

$$\begin{array}{cc}\hfill \overline{R}(e_1,e_2,e_2,e_1)& =f_1+3f_2{g}^2(\varphi e_1,e_2)-\sigma (e_2,e_2)-\sigma (e_1,e_1).\hfill \end{array}$$

(31)

We denote

$$\sigma (e_1,e_1)+\sigma (e_2,e_2)=\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)$$

and ${\psi }^2\left(\pi \right)=g^2(\varphi e_1,e_2)$. Then,

$$\begin{array}{cc}\hfill K\left(\pi \right)=& \sum _{r=1}^n[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]+f_1+3f_2{\psi }^2\left(\pi \right)-\lambda +\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).\hfill \end{array}$$

(32)

From the Equations (28) and (32), we find

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\frac{n(n+1)}{2}{f}_1+\frac{3}{2}{f}_2\sum _{i\ne j}{g}^2(\varphi e_i,e_j)-(2n-1)\lambda \hfill \\ \hfill & -n{f}_3-n{\alpha }^2{sin}^2\theta -n\sigma (e_0,e_0)\hfill \\ \hfill & +\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & -\sum _{r=1}^n[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-f_1-3f_2{\psi }^2\left(\pi \right)\hfill \\ \hfill & +\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)],\hfill \end{array}$$

(33)

or equivalently,

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\frac{(n+2)(n-1)f_1}{2}+\frac{3{cos}^2\theta -2{\psi }^2\left(\pi \right)}{2}{f}_2\hfill \\ \hfill & -n(f_3+{\alpha }^2{sin}^2\theta )-2(n-1)\lambda \hfill \\ \hfill & -n\sigma (e_0,e_0)-\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)]\hfill \\ \hfill & +\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2]-\sum _{r=1}^n[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n\{\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n[{\left(h_{1j}^r\right)}^2+{\left(h_{2j}^r\right)}^2]-\sum _{2\le i\ne j\le n}{\left(h_{ij}^r\right)}^2\}\hfill \\ \hfill & +\frac{(n+2)(n-1)}{2}{f}_1+\frac{3{cos}^2\theta -2{\psi }^2\left(\pi \right)}{2}{f}_2\hfill \\ \hfill & -n(f_3+{\alpha }^2{sin}^2\theta )-2(n-1)\lambda \hfill \\ \hfill & -n\sigma (e_0,e_0)-\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)].\hfill \end{array}$$

(34)

Because the submanifold M is a special contact slant submanifold and P is tangent to M, the components of the second fundamental satisfy $h_{ij}^k=h_{ik}^j=h_{jk}^i,\forall i,j,k=1,\cdots ,n.$

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 $2\le i\ne j\le n$.

Then, the last equation implies

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n\{\sum _{j=1}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n{\left(h_{11}^j\right)}^2-\sum _{j=3}^n{\left(h_{jj}^1\right)}^2-\sum _{2\le i\ne j\le n}{\left(h_{jj}^i\right)}^2\}\hfill \\ \hfill & +\frac{(n+2)(n-1)f_1}{2}+\frac{3{cos}^2\theta -2{\psi }^2\left(\pi \right)}{2}{f}_2\hfill \\ \hfill & -n(f_3+{\alpha }^2{sin}^2\theta )-2(n-1)\lambda \hfill \\ \hfill & -n\sigma (e_0,e_0)-\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)]\hfill \end{array}$$

(35)

For finalizing the proof, we will use some inequalities from [31].

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

(36)

for $r=1,2$, and

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

(37)

for $r=3,\cdots ,n$.

The first inequality in (36) is equivalent to

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

We have the equality sign if and only if $3h_{jj}^r=h_{11}^r+h_{22}^r,\forall j=3,\cdots ,n$.

The equality also holds in the second inequality of (36) if and only if $h_{11}^r+h_{22}^r=0$ and $h_{jj}^r=0$, $\forall j=3,\cdots ,n$, and $r=1,2$.

The inequality (37) is equivalent to

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

(38)

We have the equality if and only if $h_{11}^r=h_{22}^r=3{\lambda }^r,h_{jj}^r=4{\lambda }^r$; $\forall j=3,\cdots ,n,j\ne r,r=3,\cdots ,n$, and $h_{rr}^r=12{\lambda }^r,{\lambda }^r\in \mathbb{R}.$

By the inequalities (36) and (37), it follows that

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& \le \frac{n^2(2n-3)}{2(2n-3)}+\frac{(n+2)(n-1)}{2}{f}_1\hfill \\ \hfill & +\frac{3{cos}^2\theta -2{\psi }^2\left(\pi \right)}{2}{f}_2\hfill \\ \hfill & -n(f_3+{\alpha }^2{sin}^2\theta )-2(n-1)\lambda \hfill \\ \hfill & -n\sigma (e_0,e_0)-\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right)].\hfill \end{array}$$

(39)

The equality holds if and only if the second fundamental form takes the form as in the theorem. □

4. An Improved Chen First Inequality for Legendrian Submanifolds in Trans-Sasakian Manifolds Admitting a Semi-Symmetric Metric Connection

We establish an optimal Chen first inequaity for Legendrian submanifolds in $(\alpha ,\beta )$ trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.

Theorem 3.

Let M be an n-dimensional $(n\ge 3)$ Legendrian submanifold in a $(\alpha ,\beta )$ trans-Sasakian generalized Sasakian space form $\overline{M}(f_1,f_2,f_3)$ of dimension $2n+1$ endowed with a semi-symmetric metric connection with P tangent to M and $p\in M$, $\pi \subset T_{p}M$ a 2-plane section. Then, we have

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)\le & \frac{n^2(2n-3)}{2(2n+3)}{\parallel H\parallel }^2+\frac{(n+1)(n-2)}{2}{f}_1\hfill \\ \hfill & -n\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).\hfill \end{array}$$

(40)

Moreover, the equality case of the above inequality holds for a plane section π at $p\in M$ if and only if there exist an orthonormal basis $\{e_1,\cdots ,e_n\}$ of $T_{p}M$ with $e_1,e_2\in \pi$ such that the second fundamental form takes the following form with respect to this basis:

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

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

Proof. 

Let $p\in M$ and $\pi \subset T_{p}M$ be a 2-plane section and $\{e_1,\cdots ,e_n\}$ 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 $\{e_1^{*}=e_{n+1}=\varphi e_1,e_2^{*}=e_{n+2}=\varphi e_2\cdots e_n^{*}=e_{2n}=\varphi e_n,e_{2n+1}^{*}=\xi \}$.

In the Formula (8), we put $X=W=e_i$ and $Y=Z=e_j$. Then,

$$\begin{array}{cc}\hfill \overline{R}(e_i,e_j,e_j,e_i)& =f_1\{g(e_j,e_j)g(e_i,e_i)-g(e_i,e_j)g(e_j,e_i)\}\hfill \\ \hfill & +f_2\{g(e_i,\varphi e_j)g(\varphi e_j,e_i)-g(e_j,\varphi e_j)g(\varphi e_i,e_i)+2g(e_i,\varphi e_j)g(\varphi e_j,e_i)\}\hfill \\ \hfill & +f_3\{\eta \left(e_i\right)\eta \left(e_j\right)g(e_j,e_i)-\eta \left(e_j\right)\eta \left(e_j\right)g(e_i,e_i)\hfill \\ \hfill & +g(e_i,e_j)\eta \left(e_j\right)\eta \left(e_i\right)-g(e_j,e_j)\eta \left(e_i\right)\eta \left(e_i\right)\}\hfill \\ \hfill & -\sigma (e_j,e_j)g(e_i,e_i)+\sigma (e_i,e_j)g(e_j,e_i)-\sigma (e_i,e_i)g(e_j,e_j)+\sigma (e_j,e_i)g(e_i,e_j)\hfill \\ \hfill & =f_1-\sigma (e_j,e_j)-\sigma (e_i,e_i).\hfill \end{array}$$

(41)

In the Gauss equation with respect to the semi-symmetric metric connection we put $X=W=e_i$ and $Y=Z=e_j$.

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

(42)

By substituting the Equation (41) in the Equation (42), we have

$$\begin{array}{cc}\hfill f_1-\sigma (e_j,e_j)-\sigma (e_i,e_i)& =R(e_i,e_j,e_j,e_i)+g(h(e_i,e_j),h(e_i,e_j))\hfill \\ \hfill & -g(h(e_i,e_i),h(e_j,e_j)).\hfill \end{array}$$

(43)

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

$$\begin{array}{c}\hfill 2\tau +{\parallel h\parallel }^2-n^2{\parallel H\parallel }^2=n(n-1)f_1-2(n-1)\lambda ,\end{array}$$

(44)

where $\lambda =\mathit{trace}\sigma$.

It follows that

$$\begin{array}{c}\hfill \tau \left(p\right)=\sum _{r=1}^n\sum _{1\le i<j\le n}\left[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2\right]+\frac{n(n-1)}{2}{f}_1-(n-1)\lambda .\end{array}$$

(45)

Let $\pi$ be a 2-plane section of $T_{p}M$ at p, where $\pi =sp\{e_1,e_2\}$. By the Gauss equation, we have

$$\begin{array}{c}\hfill K\left(\pi \right)=R(e_1,e_2,e_2,e_1)=\sum _{r=1}^n[h_{11}^r{h}_{22}^r-\left(h_{12}^r\right)]+\overline{R}(e_1,e_2,e_2,e_1),\end{array}$$

(46)

or equivalently

$$\begin{array}{c}\hfill K\left(\pi \right)=f_1+\sum _{r=1}^n[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-\sigma (e_1,e_1)-\sigma (e_2,e_2).\end{array}$$

(47)

The Formulas (45) and (47) imply

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-{\left(h_{ij}^r\right)}^2-h_{11}^r{h}_{22}^r+{\left(h_{12}^r\right)}^2]\hfill \\ \hfill & +\frac{(n+1)(n-2)}{2}{f}_1-(n-1)\lambda +\sigma (e_1,e_1)+\sigma (e_2,e_2).\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n[\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n[{\left(h_{1j}^r\right)}^2+{\left(h_{2j}^r\right)}^2]-\sum _{3\le i<j\le n}{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & +\frac{(n+1)(n-2)}{2}{f}_1-(n-1)\lambda +\sigma (e_1,e_1)+\sigma (e_2,e_2).\hfill \end{array}$$

(48)

We can write

$$\sigma (e_1,e_1)+\sigma (e_2,e_2)=\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).$$

Then, the Equation (48) can be written as

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n[\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n[{\left(h_{1j}^r\right)}^2+{\left(h_{2j}^r\right)}^2]-\sum _{3\le i<j\le n}{\left(h_{ij}^r\right)}^2]\hfill \\ \hfill & +\frac{(n+1)(n-2)}{2}{f}_1-n\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).\hfill \end{array}$$

(49)

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, the above Equation (49) becomes

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)& =\sum _{r=1}^n[\sum _{j=3}^n(h_{11}^r+h_{22}^r)h_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r\hfill \\ \hfill & -\sum _{j=3}^n[{\left(h_{11}^j\right)}^2+{\left(h_{jj}^1\right)}^2]-\sum _{3\le i<j\le n}{\left(h_{jj}^i\right)}^2]\hfill \\ \hfill & +\frac{(n+1)(n-2)}{2}{f}_1-n\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).\hfill \end{array}$$

(50)

Recall the following inequalities (see [31]). For $r=1,2$,

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

(51)

The first inequality in (51) is equivalent to

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

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

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

(52)

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 case is realized if and only if

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

(53)

From the Equations (50)–(52), we obtain the desired inequality

$$\begin{array}{cc}\hfill \tau \left(p\right)-K\left(\pi \right)\le & \frac{n^2(2n-3)}{2(2n+3)}{\parallel H\parallel }^2+\frac{(n+1)(n-2)}{2}{f}_1\hfill \\ \hfill & -n\lambda -\mathit{trace}\left({\sigma }_{|{\pi }^{\perp }}\right).\hfill \end{array}$$

(54)

□

5. Conclusions

The methods described in this paper can be used for other classes of submanifolds in various ambient spaces endowed with a semi-symmetric metric connection. For example, we can combine the methods in this paper with the technics and results in [30,31,32,33,34,35,36,37,38] to obtain more interesting results relate with symmetry. In order to study corresponding problems for submanifolds in space forms endowed with a semi-symmetric non-metric connection, it is necessary to define a suitable sectional curvature. The standard definition of the sectional curvature cannot be used in this case.