Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

Abstract

In this article, we derive Chen’s inequalities involving Chen’s $\delta$-invariant ${\delta }_M$, Riemannian invariant $\delta (m_1,\cdots ,m_k)$, Ricci curvature, Riemannian invariant ${\mathsf{\Theta }}_k(2\le k\le m)$, the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms.

1. Introduction

In submanifold theory, obtaining the relationship between an intrinsic invariant and an extrinsic invariant has been the primary goal of many geometers in recent decades. Chen invariants were introduced by B.Y. Chen [1] to tackle the question raised by Chen concerning the existence of minimal immersions into a Euclidean space of arbitrary dimension [2]. Chen’s $\delta$-invariant ${\delta }_M$ of a Riemannian manifold M introduced by Chen is

$$\begin{array}{c}\hfill {\delta }_M\left(x\right)=\tau \left(x\right)-\inf \left\{K\left(\mathsf{\Pi }\right)\right|\mathsf{\Pi }\mathrm{is}\mathrm{a}\mathrm{plane}\sec \mathrm{tion}\subset T_{x}M\},\end{array}$$

(1)

where $\tau$ is the scalar curvature of M.

In [1], Chen obtained an inequality for a Riemannian submanifold $M^m$ of a real space form $\tilde{M}$ with constant sectional curvature c as

$$\begin{array}{c}\hfill {\delta }_M\le \frac{m^2(m-2)}{2(m-1)}{\Vert H\Vert }^2+\frac{1}{2}(m+1)(m-2)c,\end{array}$$

(2)

where H is the mean curvature of the submanifold $M^m$. Equation (2) is known as the first Chen inequality.

Then in [3], Chen gave the inequality for a Riemannian submanifold $M^m$ of complex-space-form ${\tilde{M}}^n\left(4c\right)$ as follows:

$$\begin{array}{c}\hfill {\delta }_M\le \frac{m^2(m-2)}{2(m-1)}{\Vert H\Vert }^2+\frac{1}{2}(m+1)(m-2)c+\frac{3}{2}{\Vert P\Vert }^{2}c-3\mathsf{\Theta }\left(\pi \right)c.\end{array}$$

(3)

Afterward, many authors obtained Chen’s inequalities for different submanifolds in various ambient spaces, such as the Kenmotsu space form [4], the Sasakian-space-form [5], the Cosympletic space form [6], the Riemannian manifold of quasi-constant curvature [7], generalized space forms [8,9], Statistical manifolds [10,11,12], quaternionic space forms [13] and the GRW spacetime [14].

Qu and Wang [15] introduced the notion of a special type of quarter-symmetric connection as a generalization of a semi-symmetric metric connection [16] and a semi-symmetric non-metric connection [17]. They studied the Einstein warped product and multiple warped products with a quarter-symmetric connection [15]. In [18], the authors obtained Chen’s inequalities for submanifolds of real space forms endowed with a quarter-symmetric connection. Mihai and $\ddot{\mathrm{O}}$zg$\ddot{\mathrm{u}}$r [19] obtained the Chen inequalities for submanifolds of complex space forms and Sasakian-space-forms with a semi-symmetric metric connection. Wang [20] obtained Chen inequalities for submanifolds of complex space forms and Sasakian-space-forms with quarter-symmetric connections which improved the results of Mihai and $\ddot{\mathrm{O}}$zg$\ddot{\mathrm{u}}$r [19]. Sular [21] obtained Chen inequalities for submanifolds of generalized space forms with a semi-symmetric metric connection. Al-Khaldi et al. [22] obtained the Chen–Ricci inequalities Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian-space-form endowed with the quarter-symmetric connection.

As a continuation of their studies, we obtained Chen inequalities for submanifolds of generalized Sasakian-space-form admitting a quarter-symmetric connection. The significance of this study is that it generalizes a large number of previously obtained results, some of which are [20,21]. The paper is organized as follows. In Section 2, we recall the properties of the quarter-symmetric connection. In Section 3, we establish the B.Y. Chen inequalities for submanifolds of a generalized Sasakian-space-form endowed with a quarter-symmetric connection. First, we prove the following inequality and also look at its equality case.

Theorem 1.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$, then

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1){\Vert {\xi }^T\Vert }^2\right)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right),\end{array}$$

where Π is a two-plane section $T_{x}M$, $x\in M$.

Next, we obtain bounds for the Riemannian invariant $\delta (m_1,\cdots ,m_k)$ and a Ricci curvature in terms of the scalar curvature of the r-plane section L, squared mean curvature and some special functions. Among others, we obtain the inequality involving the Riemannian invariant ${\mathsf{\Theta }}_k,2\le k\le m$, as follows:

$$\begin{array}{c}\hfill {\Vert H\Vert }^2\left(x\right)\ge {\mathsf{\Theta }}_k\left(x\right)-f_1-\frac{3f_2}{m(m-1)}{\Vert \mathcal{T}\Vert }^2+\frac{2f_3}{m}{\Vert {\xi }^T\Vert }^2\\ \hfill +\frac{\lambda }{m}({\psi }_1+{\psi }_2)+\frac{\mu }{m}{\psi }_2({\psi }_1-{\psi }_2)+({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right).\end{array}$$

Using Theorem 1 in Section 4, we derive Chen inequalities for the bi-slant submanifold of generalized Sasakian-space-forms.

2. Preliminaries

Suppose that ${\tilde{M}}^{m+p}$ is an $(m+p)$-dimensional Riemannian manifold with Riemannian metric g. A linear connection $\overline{\nabla }$ is known as a quarter-symmetric connection if its torsion tensor T is presented by

$$T(X_1,X_2)={\overline{\nabla }}_{X_1}{X}_2-{\overline{\nabla }}_{X_2}{X}_1-[X_1,X_2]$$

satisfies

$$T(X_1,X_2)=\mathsf{\Lambda }\left(X_2\right)\phi X_1-\mathsf{\Lambda }\left(X_1\right)\phi X_2,$$

where $\mathsf{\Lambda }$ is a 1-form, P is a vector field given by $\mathsf{\Lambda }\left(X_1\right)=g(X_1,P)$, and $\phi$ is $(1,1)$-tensor. In [15], the authors introduced a special type of quarter-symmetric connection defined as:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X_1}{X}_2={\widehat{\overline{\nabla }}}_{X_1}{X}_2+{\psi }_1\mathsf{\Lambda }\left(X_2\right)X_1-{\psi }_{2}g(X_1,X_2)P,\end{array}$$

(4)

where $\widehat{\overline{\nabla }}$ denote the Levi–Civita connection. It is easy to see that the quarter-symmetric connection $\overline{\nabla }$ includes the semi-symmetric metric connection (${\psi }_1={\psi }_2=1$) and the semi-symmetric non-metric connection (${\psi }_1=1,{\psi }_2=0$). Let the curvature tensor of $\overline{\nabla }$ be

$$\overline{R}(X_1,X_2)X_3={\overline{\nabla }}_{X_1}{\overline{\nabla }}_{X_2}{X}_3-{\overline{\nabla }}_{X_2}{\overline{\nabla }}_{X_1}{X}_3-{\overline{\nabla }}_{[X_1,X_2]}{X}_3.$$

Similarly, the curvature tensor $\widehat{\overline{R}}$ of $\widehat{\overline{\nabla }}$ can be defined as the same.

Let $M^m$ be an m-dimensional submanifold of an $(m+p)$-dimensional Riemannian manifold ${\tilde{M}}^{m+p}$ endowed with the quarter-symmetric connection $\overline{\nabla }$ and the Levi–Civita connection $\widehat{\overline{\nabla }}$. Let ∇ and $\widehat{\nabla }$ denote the induced quarter-symmetric connection and the induced Levi–Civita connection on the submanifold M. The Gauss formula with respect to ∇ and $\widehat{\nabla }$ can be presented as

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X_1}{X}_2={\nabla }_{X_1}{X}_2+h(X_1,X_2),X_1,X_2\in \mathsf{\Gamma }\left(TM\right)\\ \hfill {\widehat{\overline{\nabla }}}_{X_1}{X}_2={\widehat{\nabla }}_{X_1}{X}_2+\widehat{h}(X_1,X_2),X_1,X_2\in \mathsf{\Gamma }\left(TM\right)\end{array}$$

where h and $\widehat{h}$ are the second fundamental forms associated with the quarter-symmetric connection ∇ and the Levi-Civita connection $\widehat{\nabla }$, respectively, and are related as follows:

$$\begin{array}{c}\hfill h(X_1,X_2)=\widehat{h}(X_1,X_2)-{\psi }_{2}g(X_1,X_2)P^{\perp },\end{array}$$

(5)

where $P^{\perp }$ is the normal component of the vector field P on M. If $P^T$ represents that tangent component of the vector field P on M, then $P=P^T+P^{\perp }$.

The curvature tensor $\overline{R}$ with respect to the quarter-symmetric connection $\overline{\nabla }$ on ${\tilde{M}}^{m+p}$ can be expressed as [15]:

$$\begin{array}{c}\hfill \overline{R}(X_1,X_2,X_3,X_4)=\widehat{\overline{R}}(X_1,X_2,X_3,X_4)+{\psi }_1{\beta }_1(X_1,X_3)g(X_2,X_3)\\ \hfill -{\psi }_1{\beta }_1(X_2,X_3)g(X_1,X_4)+{\psi }_{2}g(X_1,X_3){\beta }_1(X_2,X_4)-{\psi }_{2}g(X_2,X_3){\beta }_1(X_1,X_4)\\ \hfill +{\psi }_2({\psi }_1-{\psi }_2)g(X_1,X_3){\beta }_2(X_2,X_4)-{\psi }_2({\psi }_1-{\psi }_2)g(X_2,X_3){\beta }_2(X_1,X_4),\end{array}$$

(6)

where ${\beta }_1$ and ${\beta }_2$ are symmetric $(0,2)$-tensor fields defined as

$${\beta }_1(X_1,X_2)=\left({\widehat{\overline{\nabla }}}_{X_1}\mathsf{\Lambda }\right)\left(X_2\right)-{\psi }_1\mathsf{\Lambda }\left(X_1\right)\mathsf{\Lambda }\left(X_2\right)+\frac{{\psi }_2}{2}g(X_1,X_2)\mathsf{\Lambda }\left(P\right),$$

and

$${\beta }_2(X_1,X_2)=\frac{\mathsf{\Lambda }\left(P\right)}{2}g(X_1,X_2)+\mathsf{\Lambda }\left(X_1\right)\mathsf{\Lambda }\left(X_2\right).$$

Moreover, we assume that $tr\left({\beta }_1\right)=\lambda$ and $tr\left({\beta }_2\right)=\mu$.

Suppose that R and $\widehat{R}$ are the curvature tensors of ∇ and $\widehat{\nabla }$, respectively. Then the Gauss equation with respect to the quarter-symmetric connection is as follows [15]:

$$\begin{array}{c}\hfill \overline{R}(X_1,X_2,X_3,X_4)=R(X_1,X_2,X_3,X_4)-g(h(X_1,X_4),h(X_2,X_3))\\ \hfill +g(h(X_2,X_4),h(X_1,X_3))+({\psi }_1-{\psi }_2)g(h(X_2,X_3),P)g(X_1,X_4)\\ \hfill +({\psi }_2-{\psi }_1)g(h(X_1,X_3),P)g(X_2,X_4).\end{array}$$

(7)

Let $\{e_1,\cdots ,e_m\}$ and $\{e_{m+1},\cdots ,e_{m+p}\}$ be an orthonormal frame of $T_{x}M$ and $T_x^{\perp }M$ at the point $x\in M$. Then the mean curvature vector of M associated with ∇ is $H=\frac{1}{m}{\sum }_{i=1}^{m}h(e_i,e_j)$. Similarly, the mean curvature vector of M associated to $\widehat{\nabla }$ is $\widehat{H}=\frac{1}{m}{\sum }_{i=1}^m\widehat{h}(e_i,e_j)$. In addition, the squared length of h is ${\Vert h\Vert }^2={\sum }_{i,j=1}^{m}g(h(e_i,e_j),h(e_i,e_j)$.

Now, we recall some of the Riemannian invariants introduced by Chen [23] in a Riemannian manifold. Let L be an r-dimensional subspace of $T_{x}M$, $x\in M$, $r\ge 2$ and $\{e_1,\dots ,e_r\}$ an orthonormal basis of L. The scalar curvature $\tau$ of the r-plane section L is given by

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

(8)

where $K_{ij}$ is the sectional curvature of the plane section spanned by $e_i$ and $e_j$ at $x\in M$. Suppose that $\mathsf{\Pi }\subset T_{x}M$ is a two-plane section and $K\left(\mathsf{\Pi }\right)$ is the sectional curvature of M for a plane section $\mathsf{\Pi }$ in $T_{x}M,x\in M$. Then

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

(9)

The scalar curvature $\tau \left(x\right)$ of M at the point x is presented by

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

(10)

where $\{e_1,\dots ,e_m\}$ is an orthonormal basis for $T_{x}M$.

3. B. Y. Chen Inequalities

First, we recall the well-known lemma obtained by Chen [1], which is as follows:

Lemma 1.

If $a_1,\dots ,a_m,a_{m+1}$ are $m+1$$(m\ge 2)$ real numbers such that

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^m{a}_i\right)}^2=(m-1)\left(\sum _{i=1}^m{a}_i^2+a_{m+1}\right),\end{array}$$

then $2a_1{a}_2\ge a_{m+1}$, with equality holding if and only if $a_1+a_2=a_3=\dots =a_m$.

Now, let $\tilde{M}$ be a $(2n+1)$-dimensional almost contact metric manifold with the structure $(\phi ,\eta ,g,\xi )$ where $\phi$ is a $(1,1)$-tensor, $\eta$ is a 1-form which is dual to the Reeb vector field $\xi$, and g is a Riemannian metric on $\tilde{M}$ which satisfies the follows [24]:

$$\begin{array}{c}\hfill {\phi }^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1,g(\phi X_1,\phi X_2)=g(X_1,X_2)-\eta \left(X_1\right)\eta \left(X_2\right).\end{array}$$

Because of these conditions, we have

$$\begin{array}{c}\hfill \phi \xi =0,\eta ·\phi =0,\eta \left(X_1\right)=g(X_1,\xi ),\end{array}$$

for any vector fields $X_1,X_2\in \mathsf{\Gamma }\left(T\tilde{M}\right)$.

An almost contact metric manifold $(\tilde{M},\phi ,\eta ,\xi ,g)$ whose curvature tensor satisfies

$$\begin{array}{c}\hfill \widehat{\overline{R}}(X_1,X_2)X_3=f_1\{g(X_2,X_3)X_1-g(X_1,X_3)X_2\}+f_2\{g(X_1,\phi X_3)\phi X_2\\ \hfill -g(X_2,\phi X_3)\phi X_1+2g(X_1,\phi X_2)\phi X_3\}+f_3\{\eta \left(X_1\right)\eta \left(X_3\right)X_2\\ \hfill -\eta \left(X_2\right)\eta \left(X_3\right)X_1+g(X_1,X_3)\eta \left(X_2\right)\xi -g(X_2,X_3)\eta \left(X_1\right)\xi \},\end{array}$$

(11)

for any vector field $X_1,X_2,X_3\in \mathsf{\Gamma }\left(T\tilde{M}\right)$ and $f_1,f_2,f_3$ being differentiable functions on $\tilde{M}$ is said to be a generalized Sasakian-space-form denoted by $\tilde{M}(f_1,f_2,f_3)$. The notion of a generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ was introduced by Alegre et al. [25], generalizing three important contact space forms, that is, the Sasakian-space-form $(f_1=\frac{c+3}{4},f_2=f_3=\frac{c-1}{4})$, the Kenmotsu space form $(f_1=\frac{c-3}{4},f_2=f_3=\frac{c+1}{4})$ and the Cosympletic space form $(f_1=f_2=f_3=\frac{c}{4})$.

From (6) and (11), we obtain

$$\begin{array}{c}\hfill \overline{R}(X_1,X_2,X_3,X_4)=f_1\{g(X_2,X_3)g(X_1,X_4)-g(X_1,X_3)g(X_2,X_4)\}\\ \hfill +f_2\{g(X_1,\phi X_3)g(\phi X_2,X_4)-g(X_2,\phi X_3)g(\phi X_1,X_4)\\ \hfill +2g(X_1,\phi X_2)g(\phi X_3,X_4)\}+f_3\{\eta \left(X_1\right)\eta \left(X_3\right)g(X_2,X_4)\\ \hfill -\eta \left(X_2\right)\eta \left(X_3\right)g(X_1,X_4)+g(X_1,X_3)\eta \left(X_2\right)\eta \left(X_4\right)\\ \hfill -g(X_2,X_3)\eta \left(X_1\right)\eta \left(X_4\right)\}+{\psi }_1{\beta }_1(X_1,X_3)g(X_2,X_3)\\ \hfill -{\psi }_1{\beta }_1(X_2,X_3)g(X_1,X_4)+{\psi }_{2}g(X_1,X_3){\beta }_1(X_2,X_4)\\ \hfill -{\psi }_{2}g(X_2,X_3){\beta }_1(X_1,X_4)+{\psi }_2({\psi }_1-{\psi }_2)g(X_1,X_3){\beta }_2(X_2,X_4)\\ \hfill -{\psi }_2({\psi }_1-{\psi }_2)g(X_2,X_3){\beta }_2(X_1,X_4),\end{array}$$

(12)

Let $M^m$ be a submanifold of a generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ of dimension $(2n+1)$. For any tangent vector field $X_1$ on M, we can write $\phi X_1=\mathcal{T}{X}_1+\mathcal{F}{X}_1$, where $\mathcal{T}{X}_1$ is the tangential component, and $\mathcal{F}{X}_1$ is the normal component of $\phi X_1$. The squared norm of $\mathcal{T}$ at $x\in M$ is defined as

$$\begin{array}{c}\hfill {\Vert \mathcal{T}\Vert }^2=\sum _{i,j=1}^m{g}^2(\phi e_i,e_j),\end{array}$$

(13)

where $\{e_1,\dots ,e_m\}$ is any orthonormal basis of the tangent space $T_{x}M$ and decomposing the structural vector field $\xi ={\xi }^T+{\xi }^{\perp }$, where ${\xi }^T$ and ${\xi }^{\perp }$ denotes the tangential and normal components of $\xi$. Moreover, we set ${\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)=g^2(\mathcal{T}{e}_1,e_2)=g^2(\phi e_1,e_2)$, where $\{e_1,e_2\}$ is the orthonormal basis of two-plane section $\mathsf{\Pi }$.

Theorem 2.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$, then

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1){\Vert {\xi }^T\Vert }^2\right)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right),\end{array}$$

where Π is a two-plane section $T_{x}M$, $x\in M$.

If in addition, P is a tangent vector field on $M^m$, then $H=\widehat{H}$ and the equality case holds at a point $x\in M$ if and only if there exists an orthonormal basis $\{e_1,\dots ,e_m\}$ of $T_{x}M$ and an orthonormal basis $\{e_{m+1},\dots ,e_{2n+1}\}$ of $T_x^{\perp }M$ such that the shape operators of M in $\tilde{M}(f_1,f_2,f_3)$ at x have the following forms:

$$\begin{array}{c}\hfill A_{e_{m+1}}=\left(\begin{array}{ccccc}{h}_{11}^{m+1}& 0& 0& \cdots & 0\\ 0& h_{22}^{m+1}& 0& \cdots & 0\\ 0& 0& h_{11}^{m+1}+h_{22}^{m+1}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & h_{11}^{m+1}+h_{22}^{m+1}\end{array}\right)\end{array}$$

Theorem and

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

Proof. 

Let $x\in M$ and $\{e_1,e_2,\dots ,e_m\},\{e_{m+1},\dots ,e_{2n+1}\}$ be an orthonormal basis of $T_{x}M$ and $T_x^{\perp }M$, respectively, then from (7), (10) and (12) we obtain

$$\begin{array}{c}\hfill 2\tau \left(x\right)=m^2{\Vert H\Vert }^2-{\Vert h\Vert }^2+m(m-1)f_1+3f_2{\Vert \mathcal{T}\Vert }^2-2(m-1)f_3{\Vert {\xi }^T\Vert }^2\\ \hfill -({\psi }_1+{\psi }_2)\lambda (m-1)-{\psi }_2({\psi }_1-{\psi }_2)\mu (m-1)-m(m-1)({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right).\end{array}$$

(14)

We set,

$$\begin{array}{c}\hfill c=2\tau \left(x\right)-\frac{m^2(m-2)}{m-1}{\Vert H\Vert }^2-m(m-1)f_1-3f_2{\Vert \mathcal{T}\Vert }^2+2(m-1)f_3{\Vert {\xi }^T\Vert }^2\\ \hfill +({\psi }_1+{\psi }_2)\lambda (m-1)+{\psi }_2({\psi }_1-{\psi }_2)\mu (m-1)+m(m-1)({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right),\end{array}$$

(15)

then (14) becomes

$$\begin{array}{c}\hfill m^2{\Vert H\Vert }^2=(m-1)\left({\Vert h\Vert }^2+c\right).\end{array}$$

(16)

For a chosen orthonormal basis, (16) can be written as:

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^m{h}_{ii}^{m+1}\right)}^2=(m-1)\left[\sum _{i=1}^m{\left(h_{ii}^{m+1}\right)}^2+\sum _{i\ne j}{\left(h_{ij}^{m+1}\right)}^2+\sum _{r=m+2}^{2n+1}\sum _{i,j=1}^m{\left(h_{ij}^r\right)}^2+c\right],\end{array}$$

then using Lemma 1, we have

$$\begin{array}{c}\hfill 2h_{11}^{m+1}{h}_{22}^{m+1}\ge \sum _{i\ne j}{\left(h_{ij}^{m+1}\right)}^2+\sum _{r=m+2}^{2n+1}\sum _{i,j=1}^m{\left(h_{ij}^r\right)}^2+c.\end{array}$$

(17)

Now, let $\mathsf{\Pi }=span\{e_1,e_2\}$, then from (7) and (12) we obtain

$$\begin{array}{c}\hfill R(e_1,e_2,e_2,e_1)=\sum _{r=m+1}^{2n+1}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-({\psi }_1-{\psi }_2)g(h(e_2,e_2),P)\\ \hfill +f_1+3f_2{g}^2(\phi e_1,e_2)-f_3({\eta }^2\left(e_1\right)+{\eta }^2\left(e_2\right))\\ \hfill -{\psi }_1{\beta }_1(e_2,e_2)-{\psi }_2{\beta }_1(e_1,e_1)-{\psi }_2({\psi }_1-{\psi }_2){\beta }_2(e_1,e_1).\end{array}$$

(18)

and

$$\begin{array}{c}\hfill R(e_1,e_2,e_1,e_2)=\sum _{r=m+1}^{2n+1}[{\left(h_{12}^r\right)}^2-h_{11}^r{h}_{22}^r]+({\psi }_1-{\psi }_2)g(h(e_1,e_1),P)\\ \hfill -f_1-3f_2{g}^2(\phi e_1,e_2)+f_3({\eta }^2\left(e_1\right)+{\eta }^2\left(e_2\right))\\ \hfill +{\psi }_1{\beta }_1(e_1,e_1)+{\psi }_2{\beta }_1(e_2,e_2)+{\psi }_2({\psi }_1-{\psi }_2){\beta }_2(e_2,e_2).\end{array}$$

(19)

Making use of (18) and (19) in (9), we obtain

$$\begin{array}{c}\hfill K\left(\mathsf{\Pi }\right)=\sum _{r=m+1}^{2n+1}[h_{11}^r{h}_{22}^r-{\left(h_{12}^r\right)}^2]-\frac{({\psi }_1-{\psi }_2)}{2}\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)\\ \hfill +f_1+3f_2{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)-f_3(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2)\\ \hfill -\frac{{\psi }_1}{2}tr({\beta }_1\mid \mathsf{\Pi })-\frac{{\psi }_2}{2}tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\frac{{\psi }_2}{2}({\psi }_1-{\psi }_2)tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right).\end{array}$$

(20)

Combining (14) and (20) gives

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)=(m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1){\Vert {\xi }^T\Vert }^2\right)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right)\\ \hfill +\sum _{r=m+1}^{2n+1}\left[\sum _{1\le i<j\le m}{h}_{ii}^r{h}_{jj}^r-h_{11}^r{h}_{22}^r-\sum _{1\le i<j\le m}{\left(h_{ij}^r\right)}^2+{\left(h_{12}^r\right)}^2\right].\end{array}$$

(21)

Making use of Lemma 2.4 [26], we have

$$\begin{array}{c}\hfill \sum _{r=m+1}^{2n+1}\left[\sum _{1\le i<j\le m}{h}_{ii}^r{h}_{jj}^r-h_{11}^r{h}_{22}^r-\sum _{1\le i<j\le m}{\left(h_{ij}^r\right)}^2+{\left(h_{12}^r\right)}^2\right]\le \frac{m^2(m-2)}{2(m-1)}{\Vert H\Vert }^2.\end{array}$$

(22)

In view of the last expression in (21), we obtain

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1){\Vert {\xi }^T\Vert }^2\right)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right).\end{array}$$

(23)

Now, if P is a tangent vector field on M, then (5) implies $h=\widehat{h}$ and $H=\widehat{H}$. If the equality case (23) holds at a point $x\in M$, then the equality cases of (17) and (22) hold, which gives

$$\begin{array}{c}\hfill h_{11}^{m+1}=h_{22}^{m+1}=h_{33}^{m+1}=\cdots =h_{mm}^{m+1}\\ \hfill h_{1j}^{m+1}=h_{2j}^{m+1}=0,j>2\\ \hfill h_{11}^r+h_{22}^r=0,r=m+2,\cdots ,2n+1\\ \hfill h_{ij}^r=0,i\ne j,r=m+1,\cdots ,2n+1\\ \hfill h_{ij}^{m+1}=0,i\ne j,i,j>2\end{array}$$

Therefore, choosing a suitable orthonormal basis, the shape operators take the desired forms. □

Corollary 1.

Under the same arguments as in Theorem 2,

 1. 

If the structure vector field ξ is tangent to M, we have

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1)\right)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right).\end{array}$$

 2. 

If the structure vector field ξ is normal to M, we have

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}++\frac{({\psi }_1+{\psi }_2)}{2}(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\lambda (m-1))+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right).\end{array}$$

Remark 1.

It should be noted that Theorem 2 generalizes the Theorem 6 obtained in [20]. Moreover, taking different values of $f_i,i=1,2,3$, we can obtain similar inequalities as Theorem 1 for the Kenmotsu space form and the Cosympletic space form endowed with certain types of connections by restricting the values of ${\psi }_i,i=1,2$.

Remark 2.

If in Theorem 2, we take ${\psi }_1={\psi }_2=1$ then we obtain Theorem 5.1 [21].

Corollary 2.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a semi-symmetric non-metric connection, then

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +\left({3\Vert \mathcal{T}\Vert }^2-6{\mathsf{\Theta }}^2\left(\mathsf{\Pi }\right)\right)\frac{f_2}{2}+\left(\Vert {\xi }_{\mathsf{\Pi }}{\Vert }^2-(m-1){\Vert {\xi }^T\Vert }^2\right)f_3\\ \hfill +\frac{1}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{1}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right),\end{array}$$

where Π is a two-plane section $T_{x}M$, $x\in M$.

For an integer $k\ge 0$, we denote by $S(m,k)$ the set of k-tuples $(m_1,\cdots ,m_k)$ of integers $\ge 2$ satisfying $m_1<m$ and $m_1,\cdots ,m_k\le m$. In addition, let $S\left(m\right)$ be the set of unordered k-tuples with $k\ge 0$ for a fixed m. Then, for each k-tuple $(m_1,\cdots ,m_k)\in S\left(m\right)$, Chen introduced a Riemannian invariant $\delta (m_1,\cdots ,m_k)$ as follows [23]

$$\begin{array}{c}\hfill \delta (m_1,\cdots ,m_k)\left(x\right)=\tau \left(x\right)-inf\{\tau \left(L_1\right)+\cdots +\tau \left(L_k\right)\},\end{array}$$

(24)

where $L_1,\cdots ,L_k$ run over all k mutually orthogonal subspaces of $T_{x}M$ such that dim$L_j=m_j,j\in \{1,\cdots ,k\}$. For simplicity, we set

$$\begin{array}{cc}& {\displaystyle {\mathsf{\Psi }}_1\left(L_j\right)=\sum _{1\le i<j\le r}{g}^2(\mathcal{T}{e}_i,e_j),{\mathsf{\Psi }}_2\left(L_j\right)=\sum _{1\le i<j\le r}[g{({\xi }^T,e_i)}^2+g{({\xi }^T,e_j)}^2]}\\ & {\displaystyle {\mathsf{\Psi }}_3\left(L_j\right)=\sum _{1\le i<j\le r}[{\beta }_1(e_i,e_i)+{\beta }_1(e_j,e_j)],{\mathsf{\Psi }}_4\left(L_j\right)=\sum _{1\le i<j\le r}[{\beta }_2(e_i,e_i)+{\beta }_2(e_j,e_j)]}\\ & {\displaystyle {\mathsf{\Psi }}_5\left(L_j\right)=\sum _{1\le i<j\le r}\mathsf{\Lambda }(h(e_i,e_i)+h(e_j,e_j))}\end{array}$$

As the generalization of Theorem 2, we state and prove the following results using the methods used in [26].

Theorem 3.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$, then

$$\begin{array}{c}\hfill \delta (m_1,\cdots ,m_k)\le b(m_1,\cdots ,m_k){\Vert H\Vert }^2+a(m_1,\cdots ,m_k)f_1\\ \hfill +3f_2\left(\frac{{\Vert \mathcal{T}\Vert }^2}{2}-\sum _{j=1}^k{\mathsf{\Psi }}_1\left(L_j\right)\right)-f_3\left((m-1)\Vert {\xi }^T{\Vert }^2-\sum _{j=1}^k{\mathsf{\Psi }}_2\left(L_j\right)\right)\\ \hfill -\frac{({\psi }_1+{\psi }_2)}{2}\left((m-1)\lambda -\sum _{j=1}^k{\mathsf{\Psi }}_3\left(L_j\right)\right)-\frac{{\psi }_2}{2}({\psi }_1-{\psi }_2)((m-1)\mu \\ \hfill -\sum _{j=1}^k{\mathsf{\Psi }}_4\left(L_j\right))+\frac{({\psi }_1-{\psi }_2)}{2}\left(m(m-1)\mathsf{\Lambda }\left(H\right)-\sum _{j=1}^k{\mathsf{\Psi }}_5\left(L_j\right)\right),\end{array}$$

for any k-tuples $(m_1,\cdots ,m_k)\in S\left(m\right)$. If P is a tangent vector field on M, the equality case holds at $x\in M^m$ if and only if there exists an orthonormal basis $\{e_1,\dots ,e_m\}$ of $T_{x}M$ and an orthonormal basis $\{e_{m+1},\dots ,e_{2n+1}\}$ of $T_x^{\perp }M$ such that the shape operators of M in $\tilde{M}(f_1,f_2,f_3)$ at x have the following forms:

$$\begin{array}{c}\hfill A_{e_{m+1}}=\left(\begin{array}{cccc}{a}_1& 0& \cdots & 0\\ 0& a_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_m\end{array}\right),A_{e_r}=\left(\begin{array}{cccc}{A}_1^r& \cdots & 0& 0\\ ⋮& \ddots & ⋮& ⋮\\ 0& \cdots & A_k^r& 0\\ 0& \cdots & 0& {\varsigma }_{r}I\end{array}\right),r=m+2,\cdots ,2n+1,\end{array}$$

where $a_1,\cdots ,a_m$ satisfy

$$a_1+\cdots +a_{m_1}=\cdots =a_{m_1+\cdots +m_{k-1}+1}+\cdots +a_{m_1+\cdots +m_k+1}=\cdots =a_m$$

and each $A_j^r$ is a symmetric $m_j\times m_j$ submatrix satisfying $tr\left(A_1^r\right)=\cdots =tr\left(A_k^r\right)={\varsigma }_r$, I is an identity matrix.

Proof. 

Choose an orthonormal basis $\{e_1,\dots ,e_m\}$ of $T_{x}M$ and an orthonormal basis $\{e_{m+1},\dots ,e_{2n+1}\}$ of $T_x^{\perp }M$ such that mean curvature vector H is in the direction of the normal vector to $e_{m+1}$. We set

$$\begin{array}{cc}& a_i=h_{ii}^{m+1},i=1,\cdots ,m\\ & b_1=a_1,b_2=a_2+\cdots +a_{m_1},b_3=a_{m_1+1}+\cdots +a_{m_1+m_2},\cdots ,\\ & b_{k+1}=a_{m_1+\cdots +m_{k-1}+1}+\cdots +a_{m_1+\cdots +m_{k-1}+m_k},\cdots ,b_{\gamma +1}=a_m,\end{array}$$

and consider the following sets

$$\begin{array}{c}\hfill D_1=\{1,\cdots ,m_1\},D_2=\{m_1+1,\cdots ,m_1+m_2\},\cdots ,\\ \hfill D_k=\{(m_1+\cdots +m_{k-1})+1,\cdots ,(m_1+\cdots +m_{k-1})+m_k\}.\end{array}$$

Let $L_1,\cdots ,L_k$ be a mutually orthogonal subspace of $T_{x}M$ with dim$L_j=m_j$, defined by

$$L_j=Span\{e_{m_1+\cdots +m_{j-1}+1},\cdots ,e_{m_1+\cdots +m_j}\},j=1,\cdots ,k.$$

From (7), (8) and (12), we obtain

$$\begin{array}{c}\hfill \tau \left(L_j\right)=\frac{m_j(m_j-1)}{2}{f}_1+3f_2{\mathsf{\Psi }}_1\left(L_j\right)-f_3{\mathsf{\Psi }}_2\left(L_j\right)\\ \hfill -\frac{({\psi }_1+{\psi }_2)}{2}{\mathsf{\Psi }}_3\left(L_j\right)-\frac{{\psi }_2}{2}({\psi }_1-{\psi }_2){\mathsf{\Psi }}_4\left(L_j\right)-\frac{({\psi }_1-{\psi }_2)}{2}{\mathsf{\Psi }}_5\left(L_j\right)\\ \hfill +\sum _{r=m+1}^{2n+1}\sum _{{\alpha }_j<{\beta }_j}\left[h_{{\alpha }_j{\alpha }_j}^r{h}_{{\beta }_j{\beta }_j}^r-{\left(h_{{\alpha }_j{\beta }_j}\right)}^2\right].\end{array}$$

(25)

We set

$$\begin{array}{c}\hfill \epsilon =2\tau -2b(m_1,\cdots ,m_k){\Vert H\Vert }^2-m(m-1)f_1-3f_2{\Vert \mathcal{T}\Vert }^2\\ \hfill +2(m-1)f_3{\Vert {\xi }^T\Vert }^2+({\psi }_1+{\psi }_2)\lambda (m-1)\\ \hfill +{\psi }_2({\psi }_1-{\psi }_2)\mu (m-1)+m(m-1)({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right),\end{array}$$

(26)

where

$$b(m_1,\cdots ,m_k)=\frac{{\displaystyle m^2\left(m+k-1-\sum _{j=1}^k{m}_j\right)}}{{\displaystyle 2\left(m+k-\sum _{j=1}^k{m}_j\right)}},$$

for each $(m_1,\cdots ,m_k)\in S\left(m\right)$.

In addition, let ${\displaystyle \gamma =m+k-\sum _{j=1}^k{m}_j}$. Then in view of this and (26), Equation (14) becomes

$$\begin{array}{c}\hfill m^2{\Vert H\Vert }^2={(\Vert h\Vert }^2+\epsilon )\gamma ,\end{array}$$

which can be written as

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^{\gamma +1}{b}_i\right)}^2=\gamma [\epsilon +\sum _{i=1}^{\gamma +1}{b}_i^2+\sum _{i\ne j}{\left(h_{ij}^{m+1}\right)}^2+\sum _{r=m+2}^{2n+1}\sum _{i,j=1}^m{\left(h_{ij}^r\right)}^2\\ \hfill -2\sum _{{\alpha }_1<{\beta }_1}{a}_{{\alpha }_1}{a}_{{\beta }_1}-\cdots -2\sum _{{\alpha }_k<{\beta }_k}{a}_{{\alpha }_k}{a}_{{\beta }_k}],\end{array}$$

(27)

where ${\alpha }_j,{\beta }_j\in D_j$ for all $j=1,\cdots ,k.$

Now applying Lemma 2.3 [26] in (27), we obtain

$$\begin{array}{c}\hfill \sum _{{\alpha }_1<{\beta }_1}{a}_{{\alpha }_1}{a}_{{\beta }_1}+\cdots +\sum _{{\alpha }_k<{\beta }_k}{a}_{{\alpha }_k}{a}_{{\beta }_k}\ge \\ \hfill \frac{1}{2}\left[\epsilon +\sum _{i\ne j}{\left(h_{ij}^{m+1}\right)}^2+\sum _{r=m+2}^{2n+1}\sum _{i,j=1}^m{\left(h_{ij}^r\right)}^2\right],\end{array}$$

which further implies

$$\begin{array}{c}\hfill \sum _{j=1}^k\sum _{r=m+1}^{2n+1}\sum _{{\alpha }_j<{\beta }_j}\left[h_{{\alpha }_j{\alpha }_j}^r{h}_{{\beta }_j{\beta }_j}^r-{\left(h_{{\alpha }_j{\beta }_j}^r\right)}^2\right]\ge \frac{\epsilon }{2}\\ \hfill +\frac{1}{2}\sum _{r=m+1}^{2n+1}\sum _{(\alpha ,\beta )\notin D^2}{\left(h_{\alpha \beta }^r\right)}^2+\sum _{r=m+2}^{2n+1}\sum _{{\alpha }_j\in D_j}{\left(h_{{\alpha }_j{\alpha }_j}^r\right)}^2\le \frac{\epsilon }{2},\end{array}$$

(28)

where $D^2=(D_1\times D_1)\cup \cdots \cup (D_k\times D_k)$. Combining (14), (25) and (28) gives

$$\begin{array}{c}\hfill \tau -\sum _{j=1}^k\tau \left(L_j\right)\le b(m_1,\cdots ,m_k){\Vert H\Vert }^2+a(m_1,\cdots ,m_k)f_1\\ \hfill +3f_2\left(\frac{{\Vert \mathcal{T}\Vert }^2}{2}-\sum _{j=1}^k{\mathsf{\Psi }}_1\left(L_j\right)\right)-f_3\left((m-1)\Vert {\xi }^T{\Vert }^2-\sum _{j=1}^k{\mathsf{\Psi }}_2\left(L_j\right)\right)\\ \hfill -\frac{({\psi }_1+{\psi }_2)}{2}\left((m-1)\lambda -\sum _{j=1}^k{\mathsf{\Psi }}_3\left(L_j\right)\right)-\frac{{\psi }_2}{2}({\psi }_1-{\psi }_2)((m-1)\mu \\ \hfill -\sum _{j=1}^k{\mathsf{\Psi }}_4\left(L_j\right))+\frac{({\psi }_1-{\psi }_2)}{2}\left(m(m-1)\mathsf{\Lambda }\left(H\right)-\sum _{j=1}^k{\mathsf{\Psi }}_5\left(L_j\right)\right),\end{array}$$

(29)

where, ${\displaystyle a(m_1,\cdots ,m_k)=\frac{1}{2}\left[m(m-1)-\sum _{j=1}^k{m}_j(m_j-1)\right]}$.

The equality case (29) at a point $x\in M$ holds if and only if all the previous inequalities hold; thus, the shape operators take the desired forms. □

Remark 3.

Restricting the values of $f_i,i=1,2,3$ and ${\psi }_i$ for $i=1,2$, we can obtain similar bounds as Theorem 3 for certain contact space forms endowed with certain connections.

Theorem 4.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$, then

 (i) 

For each unit vector $X_1$ in $T_{x}M$, we have

$$\begin{array}{c}\hfill Ric\left(X_1\right)\le (m-1)f_1+3f_2\sum _{j=2}^m{g}^2(\phi X_1,e_j)+f_3\left((2-m){\eta }^2\left(X_1\right)-{\Vert {\xi }^T\Vert }^2\right)\\ \hfill +[{\psi }_1+(1-m){\psi }_2]{\beta }_1(X_1,X_1)-{\psi }_1\lambda +{\psi }_2({\psi }_1-{\psi }_2)(1-m){\beta }_2(X_1,X_1)\\ \hfill -({\psi }_1-{\psi }_2)[m\mathsf{\Lambda }\left(H\right)-\mathsf{\Lambda }\left(h(X_1,X_1)\right)]+\frac{m^2}{4}{\Vert H\Vert }^2.\end{array}$$

(30)

 (ii) 

If $H\left(x\right)=0$, then a unit tangent vector $X_1$ at x satisfies the equality case of (30) if and only if $X_1\in \mathcal{M}\left(x\right)=\{X_1\in T_{x}M\mid h(X_1,X_2)=0,\forall X_2\in T_{x}M\}$.

 (iii) 

The equality of (30) holds for all unit tangent vectors at x if and only if either

 1. 

$m\ne 2,h_{ij}^r=0,i,j=1,2\dots ,m.r=m+1,\cdots ,2n+1,$ or

 2. 

$m=2,h_{11}^r=h_{22}^r,h_{12}^r=0,r=3,\cdots ,2n+1.$

Proof. 

Choosing the orthonormal basis $\{e_1,\cdots ,e_m\}$ such that $e_1=X_1$, where $X_1\in T_{x}M$ is a unit tangent vector at the point x on M. In view of (7) and (12), then proceeding similarly as the proof of Theorem 4 in [20], one can easily obtain the desire results. □

By choosing an orthonormal frame $\{e_1,\cdots ,e_k\}$ of L such that $e_1=X_1$, a unit tangent vector, Chen [23] defined the k-Ricci curvature of L at $X_1$ by

$$\begin{array}{c}\hfill Ri{c}_L\left(X_1\right)=K_{12}+K_{13}+\cdots +K_{1k}.\end{array}$$

(31)

For an integer $k,2\le k\le m$, the Riemannian invariant ${\mathsf{\Theta }}_k$ on M is defined by

$$\begin{array}{c}\hfill {\mathsf{\Theta }}_k\left(x\right)=\frac{1}{k-1}inf\{Ri{c}_L\left(X_1\right)\mid L,X_1\},x\in M\end{array}$$

where L runs over all k-plane sections in $T_{x}M$ and $X_1$ runs over all unit vectors in L. From [26], we have

$$\begin{array}{c}\hfill \tau \left(x\right)\ge \frac{m(m-1)}{2}{\mathsf{\Theta }}_k\left(x\right).\end{array}$$

(32)

Let us choose $\{e_1,\cdots ,e_m\}$ and $\{e_{m+1},\cdots ,e_{2n+1}\}$ as an orthonormal basis of $T_{x}M$ and $T_x^{\perp }M,x\in M$, respectively, where $e_{m+1}$ is parallel to the mean curvature vector H. In addition, let $\{e_1,\cdots ,e_m\}$ diagonalize the shape operator $A_{e_{m+1}}$. Then,

$$\begin{array}{c}\hfill A_{e_{m+1}}=\left(\begin{array}{cccc}{a}_1& 0& \cdots & 0\\ 0& a_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_m\end{array}\right)\end{array}$$

and

$$\begin{array}{c}\hfill A_{e_r}=h_{ij}^r,i,j=1,\cdots ,m,r=m+2,\cdots ,2n+1,tr{A}_{e_r}=0.\end{array}$$

(33)

In consequence of the above assumptions, Equation (14) can be written as follows:

$$\begin{array}{c}\hfill m^2{\Vert H\Vert }^2=2\tau +\sum _{i=1}^m{a}_i^2+\sum _{r=m+2}^{2n+1}\sum _{i,j=1}^m{\left(h_{ij}^r\right)}^2-m(m-1)f_1\\ \hfill -3f_2{\Vert \mathcal{T}\Vert }^2+2(m-1)f_3{\Vert {\xi }^T\Vert }^2+({\psi }_1+{\psi }_2)\lambda (m-1)\\ \hfill +{\psi }_2({\psi }_1-{\psi }_2)\mu (m-1)+m(m-1)({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right).\end{array}$$

(34)

Using the Cauchy–Schwartz inequality, we have

$$\begin{array}{c}\hfill \sum _{i=1}^m{a}_i^2\ge m{\Vert H\Vert }^2.\end{array}$$

(35)

Combining (32) and (34), we can state the following:

Theorem 5.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$. Then for any integer $k,2\le k\le m$ and any point $x\in M$, we have

$$\begin{array}{c}\hfill {\Vert H\Vert }^2\left(x\right)\ge {\mathsf{\Theta }}_k\left(x\right)-f_1-\frac{3f_2}{m(m-1)}{\Vert \mathcal{T}\Vert }^2+\frac{2f_3}{m}{\Vert {\xi }^T\Vert }^2\\ \hfill +\frac{\lambda }{m}({\psi }_1+{\psi }_2)+\frac{\mu }{m}{\psi }_2({\psi }_1-{\psi }_2)+({\psi }_1-{\psi }_2)\mathsf{\Lambda }\left(H\right).\end{array}$$

As a particular case of Theorem 5, we obtained Theorem 6.2 [21] which is as follows:

Corollary 3

([21]). Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a semi-symmetric metric connection. Then for any integer $k,2\le k\le m$ and any point $x\in M$, we have

$$\begin{array}{c}\hfill {\Vert H\Vert }^2\left(x\right)\ge {\mathsf{\Theta }}_k\left(x\right)-f_1-\frac{3f_2}{m(m-1)}{\Vert \mathcal{T}\Vert }^2+\frac{2f_3}{m}{\Vert {\xi }^T\Vert }^2+\frac{2\lambda }{m}.\end{array}$$

Corollary 4.

Let $M^m,m\ge 3$ be an m-dimensional submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a semi-symmetric non-metric connection. Then for any integer $k,2\le k\le m$ and any point $x\in M$, we have

$$\begin{array}{c}\hfill {\Vert H\Vert }^2\left(x\right)\ge {\mathsf{\Theta }}_k\left(x\right)-f_1-\frac{3f_2}{m(m-1)}{\Vert \mathcal{T}\Vert }^2+\frac{2f_3}{m}{\Vert {\xi }^T\Vert }^2+\frac{\lambda }{m}+\mathsf{\Lambda }\left(H\right).\end{array}$$

Remark 4.

Restricting function $f_i,i=1,2,3$, we can easily obtain similar inequality in the case of the Sasakian, Kenmotsu and Cosympletic space forms.

4. Some Applications

The notion of slant submanifolds in almost contact geometry was introduced by Lotta [27]. A submanifold M of an almost contact metric manifold $(\tilde{M},\phi ,\xi ,\eta ,g)$ tangent to the structure vector field $\xi$ is said to be a contact slant submanifold if, for any point $x\in M$ and any vector $X_1\in T_{x}M$ linearly independent on ${\xi }_x$, the angle between the vector $\phi X_1$ and the tangent space $T_{x}M$ is constant. This angle is known as the slant angle of M. The concept of slant submanifold is further generalized as follows:

Definition 1

([28]). A submanifold M of an almost contact metric manifold M is called a bi-slant submanifold, whenever we have

 1. 

$TM={\mathcal{D}}_{{\theta }_1}\oplus {\mathcal{D}}_{{\theta }_2}\oplus \xi$

 2. 

$\phi {\mathcal{D}}_{{\theta }_1}\perp {\mathcal{D}}_{{\theta }_2}$ and $\phi {\mathcal{D}}_{{\theta }_2}\perp {\mathcal{D}}_{{\theta }_1}$.

 3. 

For $i=1,2,$ the distribution $D_i$ is slant with slant angle ${\theta }_i$.

Now, as a consequence of Theorem 2, we can state the following:

Theorem 6.

Let M be a $(m=2d_1+2d_2+1)$-dimensional bi-slant submanifold of a $(2n+1)$-dimensional generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ endowed with a quarter-symmetric connection $\overline{\nabla }$, then we have

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +3\left((d_1-1)co{s}^2{\theta }_1+d_{2}co{s}^2{\theta }_2\right)\frac{f_2}{2}-(m-1)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right),\end{array}$$

for any plane Π invariant by $\mathcal{T}$ and tangent to slant distribution ${\mathcal{D}}_{{\theta }_1}$ and

$$\begin{array}{c}\hfill \tau \left(x\right)-K\left(\mathsf{\Pi }\right)\le (m-2)\left(\frac{m^2}{2(m-1)}{\Vert H\Vert }^2+(m+1)\frac{f_1}{2}\right)\\ \hfill +3\left(d_{1}co{s}^2{\theta }_1+(d_2-1)co{s}^2{\theta }_2\right)\frac{f_2}{2}-(m-1)f_3\\ \hfill +\frac{({\psi }_1+{\psi }_2)}{2}\left(tr\left({\beta }_1{\mid }_{\mathsf{\Pi }}\right)-\lambda (m-1)\right)+\frac{{\psi }_2({\psi }_1-{\psi }_2)}{2}(tr\left({\beta }_2{\mid }_{\mathsf{\Pi }}\right)\\ \hfill -\mu (m-1))+\frac{({\psi }_1-{\psi }_2)}{2}\left(\mathsf{\Lambda }\left(tr\left(h{\mid }_{\mathsf{\Pi }}\right)\right)-m(m-1)\mathsf{\Lambda }\left(H\right)\right),\end{array}$$

for any plane Π invariant by $\mathcal{T}$ and tangent to slant distribution ${\mathcal{D}}_{{\theta }_2}$. Moreover, the ideal case is the same as Theorem 2.

Proof. 

Let M be a bi-slant submanifold of a generalized Sasakian-space-form $\tilde{M}(f_1,f_2,f_3)$ of dimension $(m=2d_1+2d_2+1)$ and let $\{e_1,\cdots ,e_m=\xi \}$ be an orthonormal frame of tangent space $T_{x}M$ at a point $x\in M$, such that

$$\begin{array}{c}\hfill e_1,e_2=sec{\theta }_1\mathcal{T}{e}_1,\cdots ,e_{2d_1-1},e_{2d_1}=sec{\theta }_1\mathcal{T}{e}_{2d_1-1},e_{2d_1+1},e_{2d_1+2}\\ \hfill =sec{\theta }_2\mathcal{T}{e}_{2d_1+1},\cdots ,e_{2d_1+2d_2-1},e_{2d_1+2d_2}=sec{\theta }_2\mathcal{T}{e}_{2d_1+2d_2-1},e_{2d_1+2d_2+1}=\xi ,\end{array}$$

which gives

$$\begin{array}{c}\hfill g^2(\phi e_{i+1},e_i)=\left\{\begin{array}{c}co{s}^2{\theta }_1,\mathrm{for}i=1,2,\cdots ,2d_1-1\hfill \\ co{s}^2{\theta }_2,\mathrm{for}i=2d_1+1,\cdots ,2d_1+2d_2-1.\hfill \end{array}\right.\end{array}$$

Thus we have

$${\Vert \mathcal{T}\Vert }^2=2\{d_{1}co{s}^2{\theta }_1+d_{2}co{s}^2{\theta }_2\}$$

Making use of the above facts in Theorem 2, the proof is straightforward. □

In a similar manner, Theorems 3, 4 and 5 can be stated for a bi-slant submanifold of a generalized Sasakian-space-form. Moreover, restricting the values of ${\theta }_i,i=1,2$, similar results can be obtained for a large class of submanifolds such as slant, semi-slant, hemi-slant, semi-invariant submanifolds. Moreover, by taking different values of $f_i,i=1,2,3$, we can derive similar inequalities for the Sasakian, Kenmotsu and Cosympletic space forms.

5. Conclusions and Future Work

In this article, we established the general form of Chen’s inequalities are obtained for generalized Sasakian-space-forms endowed with a special type of quarter-symmetric connection. This work is in continuation of the previous works by Wang [20], Mihai and $\ddot{\mathrm{O}}$zg$\ddot{\mathrm{u}}$r [19], Sular [21] and Wang and Zhang [18]. By using the obtained inequality, we derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms. Recently, Chen inequality for lightlike hypersurfaces of GRW spacetime was obtained by Poyraz [14]. For future research, we would try to combine the methods and results in [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] to obtain the Chen inequalities for submanifolds of indefinite space forms such as spacelike and lightlike.