Some Pinching Results for Bi-Slant Submanifolds in S-Space Forms

Abstract

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized $\delta$-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively.

1. Introduction

In 1890, Casorati [1] pioneered the use of the Casorati curvature in place of the typical Gauss curvature. Many researchers have debated the geometrical significance of the Casorati curvature [2,3,4,5]. This curvature caught the attention of the researchers to develop optimal inequalities for Casorati curvatures ${\delta }_c$ and ${\widehat{\delta }}_c$ for different submanifolds because of its large geometric relevance [6,7,8,9,10,11,12,13]. Furthermore, some recent related studies can be seen in [14,15].

On the other hand, in 1979, Wintgen [16] proved an inequality involving the Gauss curvature K, the normal curvature $K^{\perp }$ and the squared norm of mean curvature ${\left|\right|H\left|\right|}^2$ for a surface in ${\mathbb{E}}^4$:

$$\begin{array}{c}\hfill {\left|\right|H\left|\right|}^2\ge K+\left|K^{\perp }\right|.\end{array}$$

(1)

Equation (1) is known as the Wintgen inequality. In 1983, Guadalupe et al. [17] extended the inequality for arbitrary codimension. Later, De Smet et al. [18] conjectured a generalized Wintgen inequality for any submanifold in real space forms. This conjecture is also known as the DDVV conjecture and it was proved by Ge and Tang [19]. Two of the present authors proved versions of the generalized Wintgen inequality (see [20,21,22]).

Furthermore, we note that many optimal inequalities between intrinsic and extrinsic invariants for n-dimensional Riemannian manifolds isometrically immersed in m-dimensional real space forms have been obtained by Chen [23]. In particular, these inequalities provide a lower bound for the squared mean curvature in terms of the scalar curvature. The equality holds if and only if the second fundamental form has a specific expression with respect to suitable orthonormal bases. Ideal submanifolds are submanifolds that satisfy the equality condition [23].

On the other hand, a Chen submanifold is an n-dimensional submanifold of an m-dimensional Riemannian manifold whose allied mean curvature vector [24]

$$\chi \left(H\right)=\frac{1}{n}\sum _{a=2}^{m-n}Trace\left(S_H{S}_a\right){\xi }_a$$

vanishes identically, where $\{{\xi }_1,{\xi }_2,\dots ,{\xi }_{m-n}\}$ is an orthonormal frame in the normal bundle with ${\xi }_1$ parallel to H, and $S_H$ is the shape operator at H. Any submanifold is a Chen submanifold if and only if

$$\begin{array}{c}\hfill \sum _{ı,j=1}^n\mathit{g}(\zeta (E_{ı},E_j),H)\zeta (E_{ı},E_j)\end{array}$$

is parallel to the mean curvature vector H, where $\zeta$ is the second fundamental form, and $\{E_1,\dots ,E_n\}$ an orthonormal frame.

The present article is devoted to construct some inequalities for bi-slant submanifolds in S-space forms. Thus, Section 2 and Section 3 give basic formulae and definitions related to the structure under discussion. In Section 4, we obtain optimal inequalities for the generalized normalized $\delta$-Casorati curvatures on bi-slant submanifolds in S-space forms. Also, we discuss the equality cases. Some immediate applications are also given. Section 5 deals with the study of the generalized Wintgen inequality. In this section, we derive the generalized Wintgen inequality (DDVV) for Legendrian submanifolds in S-space forms and the equality case is also investigated. Furthermore, we discuss some consequences of the derived inequality. In Section 6, we establish such an inequality for bi-slant submanifolds in the same ambient space and give some geometric applications. Finally, in Section 7, we construct some examples of slant submanifolds of the S-manifold ${\mathbb{R}}^{4+p}$ with its usual S-structure. We also provide an example supporting Corollary 3.

2. Preliminaries

Blair initiated the study of S-manifolds, which in particular reduces to Sasakian manifolds and Kaehler manifolds [25].

A $(2n+p)$-dimensional Riemannian manifold $(\overline{N},g)$ is said to be an S-manifold if it admits an endomorphism $\varphi$ of the tangent bundle of rank $2n$ and p global vector fields ${\xi }_1,\dots ,{\xi }_p$ (called structure vector fields) such that, if ${\eta }_1,\dots ,{\eta }_p$ are the dual 1-forms of ${\xi }_1,\dots ,{\xi }_p$, then

$$\varphi {\xi }_{\alpha }=0,{\eta }_{\alpha }\circ \varphi =0,{\varphi }^2=-I+\sum _{\alpha =1}^p{\eta }_{\alpha }\otimes {\xi }_{\alpha },$$

$$g(X,Y)=g(\varphi X,\varphi Y)+\sum _{\alpha =1}^p{\eta }_{\alpha }\left(X\right){\eta }_{\alpha }\left(Y\right),\forall X,Y\in \mathsf{\Gamma }\left(T\overline{N}\right),$$

$$[\varphi ,\varphi ]+2\sum _{\alpha =1}^{p}d{\eta }_{\alpha }\otimes {\xi }_{\alpha }=0,$$

$${\eta }_1\wedge \dots \wedge {\eta }_p\wedge {\left(d{\eta }_{\alpha }\right)}^n\ne 0,d{\eta }_{\alpha }=F,\forall \alpha =1,\dots ,p,$$

where $[\varphi ,\varphi ]$ is the Nijenhuis tensor of $\varphi$ and F the fundamental 2-form, defined by $F(X,Y)=g(X,\varphi Y)$.

The Levi-Civita connection $\tilde{\nabla }$ of an S-manifold satisfies

$${\tilde{\nabla }}_X{\xi }_{\alpha }=-\varphi X,$$

$$\left({\tilde{\nabla }}_X\varphi \right)Y=\sum _{\alpha =1}^p[g(\varphi X,\varphi Y){\xi }_{\alpha }+{\eta }_{\alpha }\left(Y\right){\varphi }^{2}X],$$

for any $X,Y\in \mathsf{\Gamma }\left(T\overline{N}\right)$ and any $\alpha =1,\dots ,p$.

An S-space form is an S-manifold $\overline{N}$ with constant $\varphi$-sectional curvature c, and it is denoted by $\overline{N}\left(c\right)$. The curvature tensor R of any submanifold N of $\overline{N}\left(c\right)$ is given by (see [26]):

$$\begin{array}{ccc}\hfill & & R(X,Y,Z,W)=g\left(R\right(X,Y)W,Z)\hfill \\ & =& \sum _{ı,j}\{\mathit{g}(\varphi X,\varphi W){\eta }_{ı}\left(Y\right){\eta }_j\left(Z\right)-\mathit{g}(\varphi X,\varphi Z){\eta }_{ı}\left(Y\right){\eta }_j\left(W\right)\hfill \\ & +& \mathit{g}(\varphi Y,\varphi Z){\eta }_{ı}\left(X\right){\eta }_j\left(W\right)-\mathit{g}(\varphi Y,\varphi W){\eta }_{ı}\left(X\right){\eta }_j\left(Z\right)\}\hfill \\ & +& \frac{c+3p}{4}\sum _{ı,j}\left\{\mathit{g}(\varphi X,\varphi W)\mathit{g}(\varphi Y,\varphi Z)-\mathit{g}(\varphi X,\varphi Z)\mathit{g}(\varphi Y,\varphi W)\right\}\hfill \\ & +& \frac{c-p}{4}\sum _{ı,j}\{\mathit{g}(X,\varphi W)\mathit{g}(Y,\varphi Z)-\mathit{g}(X,\varphi Z)\mathit{g}(Y,\varphi W)\hfill \\ & -& 2\mathit{g}(X,\varphi Y)\mathit{g}(Z,\varphi W)\}+\mathit{g}(\zeta (X,W),\zeta (Y,Z))\hfill \\ & -& \mathit{g}\left(\zeta \right(X,Z),\zeta (Y,W\left)\right),\hfill \end{array}$$

(2)

for any $X,Y,Z,W\in \mathsf{\Gamma }\left(TN\right)$, where $\zeta$ is the second fundamental form of N.

For any $X\in \mathsf{\Gamma }\left(TN\right)$, one decomposes $\varphi X=PX+FX,$ where $PX=tan\left(\varphi X\right)$ and $FX=nor\left(\varphi X\right)$.

Slant submanifolds in complex manifolds were defined by Chen as a natural generalization of both holomorphic and totally real immersions (see [27]). After that, Lotta [28] introduced the notion of slant immersion of a Riemannian manifold into an almost contact metric manifold.

These definitions were extended to submanifolds in S-manifolds by Carriazo et al. [29].

Definition 1

([29]). Let $\overline{N}$ be an S-manifold and N a submanifold of $\overline{N}$ tangent to all structure vector fields ${\xi }_1,\dots {\xi }_p$. N is said to be a slant submanifold if for any $\wp \in N$, and any $X\in T_{\wp }N$, linearly independent on ${\xi }_1,\dots ,{\xi }_p$, the angle θ between $\varphi X$ and $T_{\wp }N$ is constant. The angle $\theta \in [0,\frac{\pi }{2}]$ is called the slant angle of N in $\overline{N}$.

The notion of bi-slant submanifolds in almost contact metric manifolds was defined by Cabrerizo et al. [30]. It can be naturally extended to submanifolds in S-manifolds.

Definition 2.

Let $\overline{N}$ be an S-manifold and N be a submanifold in $\overline{N}$ tangent to ${\xi }_1,\dots ,{\xi }_p$. Denote by $<\xi >$ the p-dimensional distribution spanned by ${\xi }_1,\dots ,{\xi }_p$. N is said to be a bi-slant submanifold if there exist two orthogonal distributions ${\mathit{D}}_1$ and ${\mathit{D}}_2$, such that

(i) $TN$ admits the orthogonal direct decomposition $TN={\mathit{D}}_1\oplus {\mathit{D}}_2\oplus <\xi >$.

(ii) ${\mathit{D}}_{ı}$ is a slant distribution with slant angle ${\theta }_{ı}$, for any $ı=1,2$.

Particular cases of bi-slant submanifolds are semi-slant submanifolds, pseudo-slant submanifolds, CR-submanifolds and slant submanifolds. In the Table 1 below, we describe these notions [8].

Table 1. Definition.

S.N.	$\overline{\mathit{N}}$	N	${\mathit{D}}_1$	${\mathit{D}}_2$	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$
(1)	$\overline{N}$	bi-slant	slant	slant	slant angle	slant angle
(2)	$\overline{N}$	semi-slant	invariant	slant	0	slant angle
(3)	$\overline{N}$	pseudo-slant	slant	anti-invariant	slant angle	$\frac{\pi }{2}$
(4)	$\overline{N}$	CR	invariant	anti-invariant	0	$\frac{\pi }{2}$
(5)	$\overline{N}$	slant	either ${\mathit{D}}_1=0$ or ${\mathit{D}}_2=0$		either ${\theta }_1={\theta }_2=\theta$ or ${\theta }_1={\theta }_2\ne \theta$

Let $\overline{N}$ be a $(2n+p)$-dimensional S-manifold and N be a $(m+p)$-dimensional bi-slant submanifold in $\overline{N}$. Then, for orthonormal basis $\{E_1,\dots ,E_{m+p}\}$ of the tangent space $T_{\wp }N$, the squared norm of P at $\wp \in N$ is given by

$$\begin{array}{c}\hfill {\parallel P\parallel }^2=\sum _{ı,j=1}^{m+p}{\mathit{g}}^2(P{E}_{ı},E_j)=2({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2),\end{array}$$

(3)

where $2d_1$ and $2d_2$ are the dimensions of $D_1$ and $D_2$, respectively.

3. Casorati Curvatures for Submanifolds in S-Space Forms

Let $\overline{N}$ be a $(2n+p)$-dimensional S-manifold and N be an $(m+p)$-dimensional submanifold in $\overline{N}$. Let $\{E_1,\dots ,E_{m+p}\}$ be an orthonormal basis of $T_{\wp }N$ and $\{E_{m+p+1},\dots ,E_{2n+p}\}$ be an orthonormal basis of $T_{\wp }^{\perp }N$ at any $\wp \in N$. Then, the scalar curvature $\sigma (\wp )$ at ℘ is given by

$$\begin{array}{c}\hfill \sigma (\wp )=\sum _{1\le ı<j\le m+p}K(E_{ı}\wedge E_j)\end{array}$$

and the normalized scalar curvature $\rho$ is given by

$$\begin{array}{c}\hfill \rho =\frac{2\sigma }{(m+p)(m+p-1)},\end{array}$$

(4)

where $K(E_i\wedge E_j)$ denotes the sectional curvature of the plane section spanned by $E_i$ and $E_j$, i.e., $K(E_i\wedge E_j)=R(E_i,E_j,E_i,E_j)$.

The mean curvature vector H is defined as

$$\begin{array}{c}\hfill H=\frac{1}{m+p}\sum _{ı,j=1}^{m+p}\zeta (E_{ı},E_{ı})\end{array}$$

and the squared mean curvature is given by

$$\begin{array}{c}\hfill {\parallel H\parallel }^2=\frac{1}{{(m+p)}^2}\sum _{a=m+p+1}^{2n+p}{\left(\sum _{ı=1}^{m+p}{\zeta }_{ıı}^a\right)}^2,\end{array}$$

(5)

where one denotes

$${\zeta }_{ıj}^a=\mathit{g}(\zeta (E_{ı},E_j),E_a),ı,j\in \{1,2,\cdots ,m+p\},a\in \{m+p+1,\dots ,2n+p\}.$$

The Casorati curvature C of N is defined by

$$\begin{array}{c}\hfill C=\frac{1}{m+p}\sum _{a=m+p+1}^{2n+p}\sum _{ı,j=1}^{m+p}{\left({\zeta }_{ıj}^a\right)}^2.\end{array}$$

(6)

Let $\mathsf{\Gamma }$ be an r-dimensional subspace $\mathsf{\Gamma }$ of $T_{\wp }N$, $r\ge 2$, whose orthonormal basis is $\{E_1,E_2,\dots ,E_r\}$. Then, we have

$$\begin{array}{c}\hfill \sigma \left(\mathsf{\Gamma }\right)=\sum _{1\le \alpha <\beta \le r}K(E_{\alpha }\wedge E_{\beta })\end{array}$$

and

$$\begin{array}{c}\hfill C\left(\mathsf{\Gamma }\right)=\frac{1}{r}\sum _{a=m+p+1}^{2n+p}\sum _{ı,j=1}^r{\left({\zeta }_{ıj}^a\right)}^2,\end{array}$$

where $\sigma \left(\mathsf{\Gamma }\right)$ and $C\left(\mathsf{\Gamma }\right)$ are the scalar curvature and the Casorati curvature of $\mathsf{\Gamma }$, respectively.

The following $\delta$-Casorati curvatures ${\delta }_c(m+p-1)$ and ${\widehat{\delta }}_c(m+p-1)$

$$\begin{array}{c}\hfill {\left[{\delta }_c(m+p-1)\right]}_{\wp }=\frac{1}{2}{C}_{\wp }+\frac{m+p+1}{2(m+p)}inf\left\{C\left(\mathsf{\Gamma }\right)\right|\mathsf{\Gamma }:\mathrm{a}\mathrm{hyperplane}\mathrm{of} T_{\wp }N\}\end{array}$$

and

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_c(m+p-1)\right]}_{\wp }=2C_{\wp }+\frac{2(m+p)-1}{2(m+p)}sup\left\{C\left(\mathsf{\Gamma }\right)\right|\mathsf{\Gamma }:\mathrm{a}\mathrm{hyperplane}\mathrm{of} T_{\wp }N\}\end{array}$$

are known as the normalized δ-Casorati curvatures.

Furthermore, we put

$$\begin{array}{c}\hfill q\left(t\right)=\frac{1}{(m+p)t}(m+p-1)(m+p+t)[{(m+p)}^2-(m+p)-t],\end{array}$$

where $t\ne (m+p)(m+p-1)$ is a positive real number.

The following $\delta$-Casorati curvatures ${\delta }_c(t;m+p-1)$ and ${\widehat{\delta }}_c(t;m+p-1)$

$$\begin{array}{c}\hfill {\left[{\delta }_c(t;m+p-1)\right]}_{\wp }=t{C}_{\wp }+q\left(t\right)inf\left\{C\left(\mathsf{\Gamma }\right)\right|\mathsf{\Gamma }:\mathrm{a}\mathrm{hyperplane}\mathrm{of} T_{\wp }N\},\end{array}$$

for $0<t<{(m+p)}^2-(m+p)$, and

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_c(t;m+p-1)\right]}_{\wp }=t{C}_{\wp }+q\left(t\right)sup\left\{C\left(\mathsf{\Gamma }\right)\right|\mathsf{\Gamma }:\mathrm{a}\mathrm{hyperplane}\mathrm{of} T_{\wp }N\},\end{array}$$

for $t>{(m+p)}^2-(m+p)$ are called the generalized normalized δ-Casorati curvatures at ℘.

4. Bounds for Normalized Scalar Curvature in Terms of Generalized Normalized $\delta$-Casorati Curvatures

We state the following inequalities.

Theorem 1.

Let N be an $(m+p)$-dimensional bi-slant submanifold of a $(2n+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, we have:

(i) 

The generalized normalized δ-Casorati curvature ${\delta }_c(t;m+p-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2),\hfill \end{array}$$

(7)

for any real number t such that $0<t<(m+p)(m+p-1)$.

(ii) 

The generalized normalized δ-Casorati curvature ${\widehat{\delta }}_c(t;m+p-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2),\hfill \end{array}$$

(8)

for any real number $t>(m+p)(m+p-1)$.

Furthermore, the equalities hold in (7) and (8), respectively, if and only if N is a totally geodesic submanifold.

Proof. 

Let $\{E_1,\dots ,E_m,E_{m+1}={\xi }_1,\dots ,E_{m+p}={\xi }_p\}$ and $\{E_{m+p+1},\dots ,E_{2n+p}\}$ be orthonormal bases of $T_{\wp }N$ and $T_{\wp }^{\perp }N$ respectively at any point $\wp \in N$. Then, from (2), (3), (5) and (6) we have

$$\begin{array}{ccc}\hfill 2\sigma & =& {(m+p)}^2{\parallel H\parallel }^2-(m+p)C+m(m-1)\frac{c+3p}{4}+2mp\hfill \\ & +& \frac{3(c-p)}{4}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

(9)

We define a quadratic polynomial $\mathsf{\Theta }$ in terms of the components of the second fundamental form as follows:

$$\begin{array}{ccc}\hfill \mathsf{\Theta }& =& tC+q\left(t\right)C\left(\mathsf{\Gamma }\right)-2\sigma +n(m-1)\frac{c+3p}{4}+2mp\hfill \\ & +& \frac{3(c-p)}{4}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2),\hfill \end{array}$$

(10)

where $\mathsf{\Gamma }$ is a hyperplane of $T_{\wp }N$.

If we assume that $\mathsf{\Gamma }$ is spanned by $\{E_1,\dots ,E_{m+p-1}\}$, then the Equation (10) gives

$$\begin{array}{ccc}\hfill \mathsf{\Theta }& =& \frac{m+p+t}{m+p}\sum _{a=m+p+1}^{2n+p}\sum _{ı,j=1}^{m+p}{\left({\zeta }_{ıj}^a\right)}^2+\frac{q\left(t\right)}{m+p-1}\sum _{a=m+p+1}^{2n+p}\sum _{ı,j=1}^{m+p-1}{\left({\zeta }_{ıj}^a\right)}^2\hfill \\ & -& \sum _{a=m+p+1}^{2n+p}{\left(\sum _{ı=1}^{m+p}{\zeta }_{ıı}^a\right)}^2.\hfill \end{array}$$

We rewrite the above equality as follows:

$$\begin{array}{ccc}\hfill \mathsf{\Theta }& =& \sum _{a=m+p+1}^{2n+p}\sum _{ı=1}^{m+p-1}\left[\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right){\left({\zeta }_{ıı}^a\right)}^2+\frac{2(m+p+t)}{m+p}{({\zeta }_{ı(m+p)}^a)}^2\right]\hfill \\ & +& \sum _{m+p+1}^{2n+p}[2\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right)\sum _{(ı<j)=1}^{m+p}{({\zeta }_{ıj}^a)}^2\hfill \\ & -& 2\sum _{(ı<j)=1}^{m+p}{\zeta }_{ıı}^a{\zeta }_{jj}^a+\frac{t}{m+p}{\left({\zeta }_{(m+p)(m+p)}^a\right)}^2].\hfill \end{array}$$

(11)

We remark that the solution of the following system of homogeneous equations:

$$\left\{\begin{array}{c}\frac{\partial \mathsf{\Theta }}{\partial {\zeta }_{ıı}^a}=2\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right){\zeta }_{ıı}^a-2{\sum }_{k=1}^{m+p}{\zeta }_{kk}^a=0\hfill \\ \frac{\partial \mathsf{\Theta }}{\partial {\zeta }_{(m+p)(m+p)}^a}=\frac{2t}{m+p}{\zeta }_{(m+p)(m+p)}^a-2{\sum }_{k=1}^{m+p-1}{\zeta }_{kk}^a=0\hfill \\ \frac{\partial \mathsf{\Theta }}{\partial {\zeta }_{ıj}^a}=4\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right){\zeta }_{ıj}^a=0\hfill \\ \frac{\partial \mathsf{\Theta }}{\partial {\zeta }_{ı(m+p)}^a}=4(\frac{m+p+t}{m+p}){\zeta }_{ı(m+p)}^a=0,\hfill \end{array}\right.$$

(12)

$ı,j=\{1,2,\dots ,m+p-1\},ı\ne j$ and $a\in \{m+p+1,\dots ,2n+p\}$, are the critical points

$${\zeta }^c=({\zeta }_{11}^{m+p+1},{\zeta }_{12}^{m+p+1},\dots ,{\zeta }_{(m+p)(m+p)}^{m+p+1},\dots ,{\zeta }_{11}^{2n+p},\dots ,{\zeta }_{(m+p)(m+p)}^{2n+p})$$

of $\mathsf{\Theta }$ from (11).

Hence, each solution ${\zeta }^c$ has ${\zeta }_{ıj}^a=0$ for $ı\ne j$. In addition, the determinant is zero for the first two equations in (12). Furthermore, we obtain a Hessian matrix for $\mathsf{\Theta }$ as

$$\begin{array}{c}\hfill Hess\left(\mathsf{\Theta }\right)=\left(\begin{array}{ccc}I& O& O\\ O& \mathrm{II}& O\\ O& O& \mathrm{III}\end{array}\right),\end{array}$$

where O is the matrix of the corresponding dimension with all entries zero and $I,II$ and $III$ are defined below:

$$\begin{array}{c}\hfill I=\left(\begin{array}{cccc}2\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right)-2& \cdots & -2& -2\\ -2& \cdots & -2& -2\\ ⋮& \ddots & ⋮& ⋮\\ -2& \cdots & 2\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right)-2& -2\\ -2& \cdots & -2& \frac{2t}{m+p}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill II=diag\left(4\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right),\dots ,4\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right)\right),\end{array}$$

and

$$III=diag\left(\frac{4(m+p+t)}{m+p},\dots ,\frac{4(m+p+t)}{m+p}\right).$$

Moreover, $Hess\left(\mathsf{\Theta }\right)$ has the following eigenvalues (see [11,12]):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mu }_{11}=0,{\mu }_{22}=2\left(\frac{2t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right),\hfill \\ {\mu }_{33}=\dots ={\mu }_{m+pm+p}=2\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right),\hfill \\ {\mu }_{ıj}=4\left(\frac{m+p+t}{m+p}+\frac{q\left(t\right)}{m+p-1}\right),\hfill \\ {\mu }_{ım}=\frac{4(m+p+t)}{m+p},\forall ı,j\in \{1,2,\dots ,m+p-1\},ı\ne j.\hfill \end{array}\right.\end{array}$$

(13)

However, from (13) we deduce that $\mathsf{\Theta }$ is parabolic, and it has minimum at $\mathsf{\Theta }\left({\zeta }^c\right)=0$ for the solution ${\zeta }^c$ of the system (12). Hence, $\mathsf{\Theta }\ge 0$ implies

$$\begin{array}{ccc}\hfill 2\sigma & \le & tC+q\left(t\right)C\left(\mathsf{\Gamma }\right)+n(m-1)\frac{c+3p}{4}+2mp\hfill \\ & & +\frac{3(c-p)}{4}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

(14)

Then, for all tangent hyperplanes $\mathsf{\Gamma }$ of N from (14), we obtain

$$\begin{array}{ccc}\hfill \rho & \le & \frac{t}{(m+p)(m+p-1)}C+\frac{q\left(t\right)}{(m+p)(m+p-1)}C\left(\mathsf{\Gamma }\right)\hfill \\ & +& \frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

By considering the infimum over all tangent hyperplanes $\mathsf{\Gamma }$, the result is obvious. Furthermore, the equality sign holds if and only if

$$\begin{array}{c}\hfill {\zeta }_{ıj}^a=0,\forall ı,j\in \{1,\dots ,m+p\},ı\ne j;a\in \{m+p+1,\dots ,2n+p\}\end{array}$$

(15)

and

$$\begin{array}{c}\hfill {\zeta }_{(m+p)(m+p)}^a=\frac{(m+p)(m+p-1)}{t}{\zeta }_{11}^a=\dots =\end{array}$$

(16)

$$=\frac{(m+p)(m+p-1)}{t}{\zeta }_{(m+p-1)(m+p-1)}^a,$$

for all $a\in \{m+p+1,\dots ,2n+p\}$.

On the other hand, since $E_{m+p}={\xi }_p$, one has $\zeta (E_{m+p},E_{m+p})=0$. Then, (16) implies that N is totally geodesic.

Similarly, one can obtain the geometric inequality (ii). □

Corollary 1.

Let N be an $(m+p)$-dimensional bi-slant submanifold of a $(2n+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, we have:

(i) 

The normalized δ-Casorati curvature ${\delta }_c(m+p-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_c(m+p-1)+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

(17)

In addition, the equality sign holds if and only if N is a totally geodesic submanifold.

(ii) 

The normalized δ-Casorati curvature ${\widehat{\delta }}_c(m+p-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_c(m+p-1)+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

(18)

In addition, the equality sign holds if and only if N is a totally geodesic submanifold.

Proof. 

One can easily see that

$$\begin{array}{c}\hfill {\left[{\delta }_c(\frac{(m+p)(m+p-1}{2};m+p-1)\right]}_x\end{array}$$

(19)

$$=(m+p)(m+p-1){\left[{\delta }_c(m+p-1)\right]}_x,$$

at any point $x\in N$. Therefore, putting $t=\frac{(m+p)(m+p-1)}{2}$ in (7) and taking into account (19), we have our assertion. Similarly, we obtain (ii). Furthermore, equality holds in the inequality (17) if and only if

$$\begin{array}{c}\hfill {\zeta }_{ıj}^a=0,\forall ı,j\in \{1,\dots ,m+p\},ı\ne j;a\in \{m+p+1,\dots ,2n+p\}\end{array}$$

(20)

and

$$\begin{array}{c}\hfill 2{\zeta }_{(m+p)(m+p)}^a={\zeta }_{11}^a=\dots ={\zeta }_{(m+p-1)(m+p-1)}^a,\end{array}$$

(21)

for all $a\in \{m+p+1,\dots ,2n+p\}$, while equality holds in the inequality (18) if and only if (20) holds together with

$$\begin{array}{c}\hfill {\zeta }_{(m+p)(m+p)}^a=2{\zeta }_{11}^a=\dots =2{\zeta }_{(m+p-1)(m+p-1)}^a,\end{array}$$

(22)

for all $a\in \{m+p+1,\dots ,2n+p\}$.

Because $\zeta (E_{m+p},E_{m+p})=0$, it follows that the equalities (21) and (22), respectively, imply, in both cases, that N is a totally geodesic submanifold. □

Theorem 2.

Let N be an $(m+p)$-dimensional bi-slant submanifold of a $(2n+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then we have the following Table 2 for the generalized normalized δ-Casorati curvatures:

Table 2. Casorati curvatures.

S.N.	$\overline{\mathit{N}}\left(\mathit{c}\right)$	N	Inequality
(1)	$\overline{N}\left(c\right)$	semi-slant	(i)  $$\begin{gathered} \rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_1+{\mathit{d}}_{2}co{s}^2{\theta }_2) \end{gathered}$$ (ii)  $$\begin{gathered} \rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_1+{\mathit{d}}_{2}co{s}^2{\theta }_2) \end{gathered}$$
(2)	$\overline{N}\left(c\right)$	pseudo-slant	(i)  $$\begin{gathered} \rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p)(m+p-1)}\left({\mathit{d}}_{1}co{s}^2{\theta }_1\right) \end{gathered}$$ (ii)  $$\begin{gathered} \rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p)(m+p-1)}\left({\mathit{d}}_{1}co{s}^2{\theta }_1\right) \end{gathered}$$
(3)	$\overline{N}\left(c\right)$	CR	(i)  $$\begin{gathered} \rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p)(m+p-1)}{\mathit{d}}_1 \end{gathered}$$ (ii)  $$\begin{gathered} \rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}+ \\ \frac{3(c-p)}{4(m+p)(m+p-1)}{\mathit{d}}_1 \end{gathered}$$
(4)	$\overline{N}\left(c\right)$	slant	(i)  $$\begin{gathered} \rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}+ \\ \frac{3(c-p)}{4(m+p)(m+p-1)}(m+p)co{s}^2\theta \end{gathered}$$ (ii)  $$\begin{gathered} \rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}+ \\ \frac{3(c-p)}{4(m+p)(m+p-1)}(m+p)co{s}^2\theta \end{gathered}$$
(5)	$\overline{N}\left(c\right)$	invariant	(i)  $$\begin{gathered} \rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}+ \\ \frac{3(c-p)}{4(m+p)(m+p-1)}(m+p) \end{gathered}$$ (ii)  $$\begin{gathered} \rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}+ \\ \frac{3(c-p)}{4(m+p)(m+p-1)}(m+p) \end{gathered}$$
(6)	$\overline{N}\left(c\right)$	anti-invariant	(i)  $\rho \le \frac{{\delta }_c(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}$ (ii)  $\rho \le \frac{\widehat{{\delta }_c}(t;m+p-1)}{(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}$

In addition, the equality sign holds if and only if N is a totally geodesic submanifold.

Proof. 

The first four results of Theorem 2 may be easily found using Table 1 and the results in Theorem 1, whereas the following two results of Theorem 2 can be obtained by substituting $\theta =0$ and $\theta =\frac{\pi }{2}$ in the statement on slant submanifold provided in Theorem 2 for invariant and anti-invariant submanifolds, respectively. □

5. Generalized Wintgen Inequality for Legendrian Submanifolds

Let N be an m-dimensional submanifold normal to ${\xi }_1,\dots ,{\xi }_p$ in an S-space form $\overline{N}\left(c\right)$ of dimension $2n+p$. Such a submanifold is called a C-totally real submanifold. Let $\{E_1,\dots ,E_m\}$ and $\{E_{m+1},\dots ,E_{2n},E_{2n+1}={\xi }_1,\dots ,E_{2n+p}={\xi }_p\}$ be orthonormal frames on $TN$ and in the normal bundle $T^{\perp }N$, respectively.

Following [21,22], we denote by

$$\begin{array}{c}\hfill K_N=-\frac{1}{4}\sum _{a,b=1}^{2n+p-m}Trace{[S_a,S_b]}^2,\end{array}$$

(23)

where $S_a=S_{E_{m+a}}$, $a\in \{1,\dots ,2n+p-m\}$, the scalar normal curvature of N. The normalized scalar normal curvature is given by

$${\rho }_N=\frac{2}{m(m-1)}\sqrt{K_N}.$$

In our case, $S_{{\xi }_{\alpha }}=S_{{\xi }_{\beta }}=0$. Then, we have

$$\begin{array}{ccc}\hfill K_N& =& -\frac{1}{2}\sum _{1\le a<b\le 2n+p-m}Trace{[S_a,S_b]}^2\hfill \\ & =& \sum _{1\le a<b\le 2n+p-m}\sum _{1\le ı<j\le m}{\left(\mathit{g}([S_a,S_b]E_{ı},E_j)\right)}^2.\hfill \end{array}$$

(24)

We can rewrite ${\rho }_N$ in terms of the components of the second fundamental form as follows:

$$\begin{array}{c}\hfill {\rho }_N=\frac{2}{m(m-1)}\left[\sum _{1\le a<b\le 2n+p-m}\sum _{1\le ı<j\le m}\left(\sum _{\ell =1}^m({\zeta }_{j\ell }^a{\zeta }_{ı\ell }^b-{\zeta }_{ı\ell }^a{\zeta }_{j\ell }^b)\right)\right].\end{array}$$

(25)

We prepare the following lemma for later use:

Lemma 1.

Let N be an m-dimensional C-totally real submanifold of a $(2n+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, we have the following inequality:

$$\begin{array}{c}\hfill {\rho }_N\le \rho -\frac{c+3p}{4}-{\left|\right|H\left|\right|}^2.\end{array}$$

(26)

The equality holds if and only if, with respect to suitable orthonormal frames $\{E_1,\dots ,E_m\}$ and $\{E_{m+1},\dots ,E_{2n},E_{2n+1}={\xi }_1,\dots ,E_{2n+p}={\xi }_p\}$, the shape operators $S_a=S_{E_a}$, $a\in \{m+1,\dots ,2n+p\}$, take the following forms

$$\begin{array}{c}\hfill \left.\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{m+1}& =\left(\begin{array}{ccccc}{\hslash }_1& \nu & 0& \cdots & 0\\ \nu & {\hslash }_1& 0& \cdots & 0\\ 0& 0& {\hslash }_1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\hslash }_1\end{array}\right),\hfill \\ \hfill S_{m+2}& =\left(\begin{array}{ccccc}{\hslash }_2+\nu & 0& 0& \cdots & 0\\ 0& {\hslash }_2-\nu & 0& \cdots & 0\\ 0& 0& {\hslash }_2& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\hslash }_2\end{array}\right),\hfill \\ \hfill S_{m+3}& =\left(\begin{array}{ccccc}{\hslash }_3& 0& 0& \cdots & 0\\ 0& {\hslash }_3& 0& \cdots & 0\\ 0& 0& {\hslash }_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\hslash }_3\end{array}\right),S_{m+4}=\cdots =S_{2n+p}=0.\hfill \end{array}\end{array}\right\}\end{array}$$

(27)

Proof. 

For orthonormal bases $\{E_1,\dots ,E_m\}$ and $\{E_{m+1},\dots ,E_{2n+p}\}$ of $T_{\wp }N$ and $T_{\wp }^{\perp }M$, respectively, at any point $\wp \in M$, we have

$$\begin{array}{ccc}\hfill \sigma & =& \sum _{1\le ı<j\le m}\mathit{g}(R(E_{ı},E_j)E_j,E_{ı})\hfill \\ & =& \frac{m(m-1)(c+3p)}{8}+\sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}({\zeta }_{ıı}^a{\zeta }_{jj}^a-{({\zeta }_{ıj}^a)}^2).\hfill \end{array}$$

(28)

On the other hand, we write $m^2{\left|\right|H\left|\right|}^2$ as follows:

$$\begin{array}{ccc}\hfill m^2{\left|\right|H\left|\right|}^2& =& \sum _{a=m+1}^{2n+p}{\left(\sum _{ı=1}^m{\zeta }_{ıı}^a\right)}^2\hfill \\ & =& \frac{1}{m-1}\sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}{({\zeta }_{ıı}^a-{\zeta }_{jj}^a)}^2\hfill \\ & & +\frac{2m}{m-1}\sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}{\zeta }_{ıı}^a{\zeta }_{jj}^a.\hfill \end{array}$$

(29)

Following [31], we have

$$\begin{array}{ccc}\hfill & & \sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}{({\zeta }_{ıı}^a-{\zeta }_{jj}^a)}^2+2m\sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}{\zeta }_{ıı}^a{\zeta }_{jj}^a\hfill \\ & \ge & 2m{\left\{\sum _{m+1\le a<b\le 2n+p}\sum _{1\le ı<j\le m}{\left[\sum _{\ell =1}^m({\zeta }_{j\ell }^a{\zeta }_{ı\ell }^b-{\zeta }_{ı\ell }^a{\zeta }_{j\ell }^b)\right]}^2\right\}}^{\frac{1}{2}}.\hfill \end{array}$$

(30)

Applying (30) into (29), we derive

$$\begin{array}{c}\hfill m^2{\left|\right|H\left|\right|}^2\ge \frac{2m}{m-1}{\left\{\sum _{m+1\le a<b\le 2n+p}\sum _{1\le ı<j\le m}{\left[\sum _{\ell =1}^m({\zeta }_{j\ell }^a{\zeta }_{ı\ell }^b-{\zeta }_{ı\ell }^a{\zeta }_{j\ell }^b)\right]}^2\right\}}^{\frac{1}{2}}.\end{array}$$

(31)

From (25) and (31), we easily find

$$\begin{array}{c}\hfill m^2{\left|\right|H\left|\right|}^2-m^2{\rho }_N\ge \frac{2m}{m-1}\sum _{a=m+1}^{2n+p}\sum _{1\le ı<j\le m}({\zeta }_{ıı}^a{\zeta }_{jj}^a-{({\zeta }_{ıj}^a)}^2).\end{array}$$

(32)

Substituting (32) into (28), we obtain the desired inequality.

The equality case holds in the inequality (26) if and only if the shape operators take the forms (27) with respect to the above mentioned orthonormal frames. □

An m-dimensional C-totally real submanifold in a $(2m+p)$-dimensional S-manifold is called a Legendrian submanifold.

Proposition 1.

Let N be an m-dimensional Legendrian submanifold in a $(2m+p)$-dimensional S-space form $\overline{N}\left(c\right)$. If

(i) 

the normal connection of N is flat and

(ii) 

$\rho =\frac{c+3p}{4}$,

then N is minimal.

Proof. 

From (26) of Lemma 1 and the conditions (i) and (ii), we find that ${\left|\right|H\left|\right|}^2=0$, which implies that N is a minimal submanifold in $\overline{N}\left(c\right)$. □

Next, we prove the following:

Theorem 3.

Let N be an m-dimensional Legendrian submanifold N of a $(2m+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then:

$$\begin{array}{ccc}\hfill {\left({\rho }^{\perp }\right)}^2& \le & \frac{c-p}{m(m-1)}\left(\frac{c-p}{8}+\rho -\frac{c+3p}{4}\right)+{\left(\rho -\frac{c+3p}{4}-{\left|\right|H\left|\right|}^2\right)}^2.\hfill \end{array}$$

(33)

The equality holds if and only if the shape operators take the forms (27).

Proof. 

From Gauss and Ricci equations, we find

$$\begin{array}{ccc}\hfill \mathit{g}(R^{\perp }(X,Y)V,U)& =& \frac{c-p}{4}\left(\mathit{g}(\varphi X,U)\mathit{g}(\varphi Y,V)-\mathit{g}(\varphi X,V)\mathit{g}(\varphi Y,U)\right)\hfill \\ & -& \mathit{g}([S_U,S_V]X,Y),\hfill \end{array}$$

(34)

for any $X,Y\in \mathsf{\Gamma }\left(TN\right)$ and $U,V\in \mathsf{\Gamma }\left(T^{\perp }N\right)$.

For orthonormal bases $\{E_1,\dots ,E_m\}$ and $\{E_{m+1}=\varphi E_1,\dots ,E_{2m}=\varphi E_m,E_{2m+1}={\xi }_1,\dots ,{\xi }_{2m+p}={\xi }_p\}$ of $T_{\wp }N$ and $T_{\wp }^{\perp }N$, respectively, $\wp \in N$, the Equation (34) gives

$$\begin{array}{ccc}\hfill \mathit{g}(R^{\perp }(E_{ı},E_j)E_{m+a},E_{m+b})& =& \frac{c-p}{4}(\mathit{g}(\varphi E_{ı},E_{m+b})\mathit{g}(\varphi E_j,E_{m+a})\hfill \\ & -& \mathit{g}(\varphi E_{ı},E_{m+a})\mathit{g}(\varphi E_j,E_{m+b}))-\mathit{g}([S_b,S_a]E_{ı},E_j),\hfill \end{array}$$

(35)

$\forall ı,j\in \{1,\dots ,m\}$ and $a,b\in \{m+1,\dots ,2m+p\}$. Then

$$\begin{array}{ccc}\hfill {\left({\rho }^{\perp }\right)}^2& =& {\left(\frac{2{\sigma }^{\perp }}{m(m-1)}\right)}^2\hfill \\ & =& \frac{4}{{\left(m(m-1)\right)}^2}\{\sum _{1\le a<b\le m+p}\sum _{1\le ı<j\le m}{\mathit{g}}^2(R^{\perp }(E_{ı},E_j)E_{m+a},E_{m+b})\}\hfill \\ & =& \frac{4}{{\left(m(m-1)\right)}^2}\{\sum _{1\le a<b\le m+p}\sum _{1\le ı<j\le m}[\frac{c-p}{4}(\mathit{g}(\varphi E_{ı},E_{m+b})\mathit{g}(\varphi E_j,E_{m+a})\hfill \\ & -& \mathit{g}(\varphi E_{ı},E_{m+a})\mathit{g}(\varphi E_j,E_{m+b}))+\mathit{g}([S_a,S_b]E_{ı},E_j){]}^2\}.\hfill \end{array}$$

Using basic algebraic formulas and putting (24) into the last equation, we have

$$\begin{array}{ccc}\hfill {\left({\rho }^{\perp }\right)}^2& =& \frac{4}{{\left(m(m-1)\right)}^2}\{K_N+{\left(\frac{c-p}{4}\right)}^2\left(\frac{m(m-1)}{2}\right)-\left(\frac{c-p}{4}\right){\left|\right|\zeta \left|\right|}^2\hfill \\ & +& \left(\frac{m^2(c-p)}{4}\right){\left|\right|H\left|\right|}^2\}\hfill \\ & =& \frac{4}{{\left(m(m-1)\right)}^2}\{\frac{{\left(m(m-1)\right)}^2}{4}{\rho }_N^2+\frac{m(m-1){(c-p)}^2}{32}\hfill \\ & +& \frac{c-p}{4}\left(-{\left|\right|\zeta \left|\right|}^2+m^2{\left|\right|H\left|\right|}^2\right)\}\hfill \\ & =& {\rho }_N^2+\frac{{(c-p)}^2}{8m(m-1)}+\frac{c-p}{{\left(m(m-1)\right)}^2}\left(-{\left|\right|\zeta \left|\right|}^2+m^2{\left|\right|H\left|\right|}^2\right).\hfill \end{array}$$

(36)

By using the Gauss equation, it is easy to see that

$$\begin{array}{ccc}\hfill \rho & =& \frac{2\sigma }{m(m-1)}\hfill \\ & =& \frac{1}{m(m-1)}\left\{\left(m(m-1)\right)\left(\frac{c+3p}{4}\right)+m^2{\left|\right|H\left|\right|}^2-{\left|\right|\zeta \left|\right|}^2\right\}\hfill \\ & =& \left(\frac{c+3p}{4}\right)+\frac{1}{m(m-1)}\left(-{\left|\right|\zeta \left|\right|}^2+m^2{\left|\right|H\left|\right|}^2\right).\hfill \end{array}$$

(37)

Combining (36) and (37), we obtain

$$\begin{array}{c}\hfill {\left({\rho }^{\perp }\right)}^2={\rho }_N^2+\frac{{(c-p)}^2}{8m(m-1)}+\frac{c-p}{m(m-1)}\left(\rho -\left(\frac{c+3p}{4}\right)\right).\end{array}$$

(38)

However, $K_N\ge 0$ implies ${\rho }_N\ge 0$. Therefore, (26) can be rewritten as

$$\begin{array}{c}\hfill {\rho }_N^2\le {\left(\rho -\frac{c+3p}{4}-{\left|\right|H\left|\right|}^2\right)}^2.\end{array}$$

(39)

Substituting (39) into (38), we derive

$$\begin{array}{ccc}\hfill {\left({\rho }^{\perp }\right)}^2& \le & \frac{{(c-p)}^2}{8m(m-1)}+\frac{c-p}{m(m-1)}\left(\rho -\left(\frac{c+3p}{4}\right)\right)+{\left(\rho -\frac{c+3p}{4}-{\left|\right|H\left|\right|}^2\right)}^2,\hfill \end{array}$$

which is equivalent with the inequality to prove. □

Following [32], any submanifold satisfying the equality in the Wintgen inequality is a Wintgen ideal submanifold. Then, we conclude that N is a Wintgen ideal submanifold if and only if the shape operators take the forms (27).

Some immediate consequences of Theorem 3 are the following:

Corollary 2.

Let N be an m-dimensional Wintgen ideal Legendrian submanifold in a $(2m+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, N is a Chen submanifold.

Proof. 

Following [33], we obtain that any Wintgen ideal submanifold is a Chen submanifold. □

Corollary 3.

Let N be an m-dimensional minimal Legendrian submanifold in a $(2m+p)$-dimensional S-space form $\overline{N}\left(c\right)$. If $c=\rho =p$, then the normal connection of N is flat.

Proof. 

From (33) (see Theorem 3) and our assumptions, we have ${\rho }^{\perp }=0$. This further implies that $R^{\perp }=0$, which says that the normal connection of N is flat. □

6. Generalized Wintgen Inequality for Bi-Slant Submanifolds

We state and prove the generalized Wintgen inequality for $(m+p)$-dimensional contact bi-slant submanifolds in $(2n+p)$-dimensional S-space forms.

Theorem 4.

Let N be an $(m+p)$-dimensional bi-slant submanifold of a $(2n+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, we have

$$\begin{array}{ccc}\hfill \rho +{\rho }_N& \le & {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \frac{3(c-p)}{4(m+p)(m+p-1)}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

(40)

Moreover, N is a Wintgen ideal bi-slant submanifold of $\overline{N}\left(c\right)$ if and only if, with respect to suitable orthonormal frames $\{E_1,\dots ,E_m,E_{m+1}={\xi }_1,\dots ,E_{m+p}={\xi }_p\}$ and $\{E_{m+p+1},\dots ,E_{2n+p}\}$, the shape operators $S_a=S_{E_a}$, $a\in \{m+p+1,\dots ,2n+p\}$, take the following forms

$$\begin{array}{c}\hfill \left.\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{m+p+1}& =\left(\begin{array}{ccccc}0& \nu & 0& \cdots & 0\\ \nu & 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right),\hfill \\ \hfill S_{m+p+2}& =\left(\begin{array}{ccccc}\nu & 0& 0& \cdots & 0\\ 0& -\nu & 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right),\hfill \\ \hfill S_{m+p+3}& =\cdots =S_{2n+p}=0.\hfill \end{array}\end{array}\right\}\end{array}$$

(41)

Proof. 

Let $\{E_1,\dots ,E_m,E_{m+1}={\xi }_1,\dots ,E_{m+p}={\xi }_p\}$ and $\{E_{m+p+1},\dots ,E_{2n+p}\}$ be orthonormal bases of $T_{\wp }N$ and $T_{\wp }^{\perp }N$, respectively, $\wp \in N$. Then, the Gauss equation gives

$$\begin{array}{ccc}\hfill \sigma & =& \sum _{1\le ı<j\le m+p}\mathit{g}(R(E_{ı},E_j)E_j,E_{ı})\hfill \\ & =& m(m-1)\left(\frac{c+3p}{8}\right)+mp+\left(\frac{3(c-p)}{8}\right)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2)\hfill \\ & +& \sum _{a=m+p+1}^{2n+p}\sum _{1\le ı<j\le m+p}({\zeta }_{ıı}^a{\zeta }_{jj}^a-{({\zeta }_{ıj}^a)}^2).\hfill \end{array}$$

(42)

Furthermore,

$$\begin{array}{ccc}\hfill \rho & =& \frac{2\sigma }{(m+p)(m+p-1)}\hfill \\ & =& \frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \left(\frac{3(c-p)}{4(m+p)(m+p-1)}\right)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2)\hfill \\ & +& \frac{2}{(m+p)(m+p-1)}\{\sum _{a=m+p+1}^{2n+p}\sum _{1\le ı<j\le m+p}({\zeta }_{ıı}^a{\zeta }_{jj}^a-{({\zeta }_{ıj}^a)}^2)\}.\hfill \end{array}$$

(43)

By similar arguments as in Lemma 1, we obtain

$$\begin{array}{c}\hfill \frac{(m+p)(m+p-1)}{2}\left({\left|\right|H\left|\right|}^2-{\rho }_N\right)\ge \sum _{a=m+p+1}^{2n+p}\sum _{1\le ı<j\le m+p}({\zeta }_{ıı}^a{\zeta }_{jj}^a-{({\zeta }_{ıj}^a)}^2).\end{array}$$

(44)

Substituting (44) into (43), we obtain

$$\begin{array}{ccc}\hfill \rho & \le & {\left|\right|H\left|\right|}^2-{\rho }_N+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+\frac{2mp}{(m+p)(m+p-1)}\hfill \\ & +& \left(\frac{3(c-p)}{4(m+p)(m+p-1)}\right)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2).\hfill \end{array}$$

Hence, we obtain our desired result. □

For particular cases of bi-slant submanifolds in S-space forms, one derives the following forms of the generalized Wintgen inequality (see Table 3).

Table 3. The forms of the DDVV inequality for different submanifolds.

Generalized Wintgen Inequality
S.N.	$\overline{\mathit{N}}\left(\mathit{c}\right)$	$\mathit{N}$	Inequality
(1)	$\overline{N}\left(c\right)$	pseudo-slant	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\left(\frac{3(c-p)}{4(m+p)(m+p-1)}\right){\mathit{d}}_{1}co{s}^2\theta \end{gathered}$$
(2)	$\overline{N}\left(c\right)$	semi-slant	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\left(\frac{3(c-p)}{4(m+p)(m+p-1)}\right)({\mathit{d}}_1+ \\ {\mathit{d}}_{2}co{s}^2\theta ) \end{gathered}$$
(3)	$\overline{N}\left(c\right)$	slant	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\left(\frac{3(c-p)}{4(m+p-1)}\right)co{s}^2\theta \end{gathered}$$
(4)	$\overline{N}\left(c\right)$	CR	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p-1)} \end{gathered}$$
(5)	$\overline{N}\left(c\right)$	invariant	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)}+\frac{3(c-p)}{4(m+p-1)} \end{gathered}$$
(6)	$\overline{N}\left(c\right)$	anti-invariant	$$\begin{gathered} \rho +{\rho }_N\le {\left|\right|H\left|\right|}^2+\frac{m(m-1)(c+3p)}{4(m+p)(m+p-1)}+ \\ \frac{2mp}{(m+p)(m+p-1)} \end{gathered}$$

Corollary 4.

Let N be an m-dimensional Wintgen ideal bi-slant submanifold in a $(2m+p)$-dimensional S-space form $\overline{N}\left(c\right)$. Then, N is a Chen submanifold.

7. Some Examples

In this section, we construct some examples of slant submanifolds in an S-manifold ${\mathbb{R}}^{4+s}$ endowed with the usual S-structure (inspired by Carriazo et al. [29]).

Example 1.

For any non-zero real numbers a and b,

$$\begin{array}{c}\hfill r(u,v,t_1,\dots ,t_p)=2(avsinu,avcosu,vsinbu,vcosbu,t_1,\dots ,t_p)\end{array}$$

defines a (2+p)-dimensional slant submanifold N with slant angle θ in ${\mathbb{R}}^{4+p}$.

In this case, an orthogonal frame $\{E_1,E_2,{\xi }_1,\dots ,{\xi }_p\}$ of the tangent bundle of the submanifold is given by

$$\begin{array}{cc}\hfill & E_1=\frac{\partial }{\partial u}+\sum _{\alpha =1}^p(2v^{2}asinbucosu-2v^{2}acosbusinu)\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_2=\frac{\partial }{\partial v}+\sum _{\alpha =1}^p(2vasinbucosu+2vacosbucosu)\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_{2+\alpha }=\frac{\partial }{\partial t_{\alpha }}={\xi }_{\alpha },\hfill \end{array}$$

for $\alpha =1,\dots ,p$.

Example 2.

We consider

$$\begin{array}{c}\hfill r(u,v,t_1,\dots ,t_p)=2(u,v,\kappa cosv,\kappa sinv,t_1,\dots ,t_p),\end{array}$$

for any nonzero constant κ. We choose an orthogonal basis $\{E_1,E_2,{\xi }_1,\dots ,{\xi }_p\}$ of the tangent bundle of the submanifold as follows

$$\begin{array}{cc}\hfill & E_1=\frac{\partial }{\partial u}+\sum _{\alpha =1}^p(2\kappa cosv)\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_2=\frac{\partial }{\partial v}+\sum _{\alpha =1}^p(2\kappa sinv)\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_{2+\alpha }=\frac{\partial }{\partial t_{\alpha }}={\xi }_{\alpha },\hfill \end{array}$$

for $\alpha =1,\dots ,p$. Then, N is a (2+p)-dimensional slant submanifold with slant angle $\theta =co{s}^{-1}\left(\frac{1}{\sqrt{1+{\kappa }^2}}\right)$ in ${\mathbb{R}}^{4+p}$.

Example 3.

For any nonzero constant κ,

$$\begin{array}{c}\hfill r(u,v,t_1,\dots ,t_p)=2(-\kappa vsinu,f\left(v\right),\kappa vcosu,h\left(v\right),t_1,\dots ,t_p)\end{array}$$

defines a $(2+p)$-dimensional slant submanifold N with slant angle θ in ${\mathbb{R}}^{4+p}$.

An orthogonal frame $\{E_1,E_2,{\xi }_1,\dots ,{\xi }_p\}$ of $TN$ is defined by

$$\begin{array}{cc}\hfill & E_1=\frac{\partial }{\partial u}-\sum _{\alpha =1}^p\left(2{\kappa }^2{v}^2{cos}^{2}u\right)\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_2=\frac{\partial }{\partial v}+\sum _{\alpha =1}^p(-{\kappa }^2{v}^{2}sin2u+2h\left(v\right)f^{\prime }\left(v\right))\frac{\partial }{\partial t_{\alpha }},\hfill \\ \hfill & E_{2+\alpha }=\frac{\partial }{\partial t_{\alpha }}={\xi }_{\alpha },\hfill \end{array}$$

for $\alpha =1,\dots ,p$.

Finally, we construct the following example for Corollary 3:

Example 4.

We consider a totally geodesic Legendrian immersion $i:N^2\to {\mathbb{S}}^5$ of a surface $N^2$ in the 5-dimensional unit hypersphere ${\mathbb{S}}^5$ of ${\mathbb{C}}^3$. It is known from [34] that its scalar curvature $\sigma =\frac{c+3}{2}$ and since $c=1$, it follows that the normalized scalar curvature $\rho =1$. Obviously, any totally geodesic submanifold is minimal. We observe that $c=\rho =p=1$. Then, by Corollary 3, the normal connection of N is flat.

8. Conclusions

In the present article, we established geometric inequalities for certain submanifolds in S-space forms.

The first inequality is an estimate of the Casorati curvatures of bi-slant submanifolds in terms of the scalar curvature and the following are generalized Wintgen inequalities for Legendrian submanifolds and bi-slant submanifolds, respectively. The equality cases were investigated and geometric applications were given.

The methods can be easily adapted to other ambient spaces and/or different special classes of submanifolds.