Inequalities of warped product submanifolds in a Riemannian manifold of quasi-constant curvature

1.   Introduction

In 1993, Chen^[1] introduced $\delta$-invariants, and established relationships between intrinsic invariants and extrinsic invariants for minimal submanifolds. In 1995, Chen ^[2] found Chen-like inequalities for Riemannian submanifolds and gave some applications of $\delta$-invariants. Submanifolds are ideal submanifolds when Chen-like inequalities are equal and they receive the least possible tension at each point from ambient spaces.

The Casorati curvature was originally introduced in $1980$ for surfaces in $3$-dimensional Euclidean space and is defined as the normalized square of the length of the second fundamental form (see Ref. [3]). In 2007, Decu et al. ^[4] introduced the normalized $\delta$-Casorati curvatures $\widehat{\delta_C}(n-1)$ and $\delta_C(n-1)$ and established two optimal inequalities involving the scalar curvature and the normalized $\delta$-Casorati curvature. In $2008$, Decu et al.^[5] introduced the generalized normalized $\delta$-Casorati curvatures $\widehat{\delta_C}(r;n-1)$ and $\delta_C(r;n-1)$ and proved two sharp inequalities. In 2017, Park^[6] obtained two types of optimal inequalities for the real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. In 2020, Choudhary and Blaga^[7] established some sharp inequalities involving generalized normalized $\delta$-Casorati curvatures for invariant, anti-invariant and slant submanifolds in metallic Riemannian space forms and characterized the submanifolds for which the equality holds.

In this study, we establish Chen-like inequalities for generalized normalized $\delta$-Casorati curvatures of warped product submanifolds in a Riemannian manifold of quasi-constant curvature.

Let $N_1^p$ and $N_2^q$ be two Riemannian manifolds with positive dimensions equipped with Riemannian metrics $g_{N_1^p}$ and $g_{N_2^q}$, respectively. Let $f$ be a positive function on $N_1^p$. Consider the product manifold $N_1^p{\times}N_2^q$, with its projections $\pi:N_1^p {\times} N_2^q {\rightarrow} N_1^p$ and $\eta : N_1^p {\times} N_2^q {\rightarrow} N_2^q$. The warped product manifold $M^n = N_1^p {\times}$${}_f N_2^q$ is the product manifold $N_1^p {\times} N_2^q$ equipped with a Riemannian structure such that

for any tangent vector $X {\in} T_xM^n$. Thus, we have $g = g_{N_1^p} +$$f^2g_{N_2^q}$. The function $f$ is called the warping function of the warped product manifold.

A Riemannian manifold $(\widetilde{M}^m,\tilde{g})$ is called a Riemannian manifold of quasi-constant curvature if the curvature tensor satisfies the following condition (see Ref. [8]):

where $a$ and $b$ are scalar functions, $T$ is a $1$-form defined by

where $P$ denotes the unit vector field. We uniquely decompose the vector field $P$ on $M^n$ into its tangent component $P^T$ and normal component $P^{\perp}$, that is,

Theorem 1.1. Let ${\phi}:M^n = N_1^p{\times}_fN_2^q{\rightarrow}\widetilde{M}^m$ be an isometric immersion of an $n$-dimensional warped product submanifold $M^n$ into an $m$-dimensional Riemannian manifold of a quasi-constant curvature $\widetilde{M}^m$. Then

${\rm{(i)}}$ the generalized normalized $\delta$-Casorati curvature $\delta_C(r;$$n-1)$ satisfies

for any real number $r$ such that $0<r<n(n-1)$, where ${{\parallel}P^T{\parallel}}^2_{N_1^p} = {\sum\nolimits_{i = 1}^{p}} g(P^T,e_i)^2$, ${{\parallel}P^T{\parallel}}^2_{N_2^q} = {\sum\nolimits_{s = p+1}^{n}}g(P^T,e_s)^2$, $\rho$ is the normalized scalar curvature, ${{\parallel}H{\parallel}}^2$ is the squared mean curvature, $a$ and $b$ are scalar functions;

${\rm{(ii)}}$ the generalized normalized $\delta$-Casorati curvature $\widehat{\delta_C}(r;n-1)$ satisfies

for any real number $r>n(n-1)$.

Equalities hold in (5) and (6) if and only if the shape operators for the suitable tangent and normal orthonormal frames are given by

where $f_1$ is a function on $M^n$.

Let $b = 0$ and $a = {\rm{const}}$. Then we have

Corollary 1.1. Let ${\phi}:M^n = N_1^p{\times}_fN_2^q{\rightarrow}\widetilde{M}^m(a)$ be an isometric immersion of an $n$-dimensional warped product submanifold $M^n$ into an $m$-dimensional Riemannian manifold of a constant sectional curvature $a$. Then

${\rm{(i)}}$ the generalized normalized $\delta$-Casorati curvature $\delta_C(r;$$n-1)$ satisfies

for any real number $r$ such that $0<r<n(n-1)$;

${\rm{(ii)}}$ the generalized normalized $\delta$-Casorati curvature $\widehat{\delta_C}(r;$$n-1)$ satisfies

for any real number $r>n(n-1)$.

Equalities hold in (8) and (9) if and only if the shape operators for the suitable tangent and normal orthonormal frames are given by Eq. (7).

Moreover, let $p = 0,q = n$ and $f = 1$. Then we have

Corollary 1.2. Let ${\phi}:M^n{\rightarrow}\widetilde{M}^m(a)$ be an isometric immersion of an $n$-dimensional warped product submanifold into $\widetilde{M}^m(a)$. We have

${\rm{(i)}}$ for any real number $r$ such that $0 < r < n(n-1)$,

${\rm{(ii)}}$ for any real number $r$ such that $r>n(n-1)$,

Equalities hold in (10) and (11) if and only if $M^n$ is an invariantly quasi-umbilical submanifold.

Remark: Corollary 1.2 is Theorem 2.1, and Corollary 3.1 in Ref. [5].

2.   Preliminaries

Let $M^n$ be an $n$-dimensional warped product submanifold of an $m$-dimensional Riemannian manifold of quasi-constant curvature ${\widetilde{M}}^m$. Let $\nabla$ and $\widetilde{\nabla}$ be the ${\rm{Levi}}$–${\rm{Civita}}$ connection on $M^n$ and $\widetilde{M}^m$, respectively. Then, the Gauss and Weingarten formulas are given respectively by

for vector fields $X$, $Y$ tangent to $M^n$, and vector field $N$ normal to $M^n$. Here $h$ denotes the second fundamental form, $\nabla^{\perp}$ is the normal connection and $A$ is the shape operator. The second fundamental form and shape operator are related by

where $\tilde{g}$ and $g$ denote the metric on $\widetilde{M}^m$ and $M^n$ respectively. If $R$ and $\widetilde{R}$ are the curvature tensors of $M^n$ and ${\widetilde{M}}^m$, respectively, then the Gauss equation is given by

for any vector field $X$, $Y$, $Z$, and $W$ tangent to $M^n$.

Let $\{e_1,\cdots,e_n\}$ be an orthonormal basis of the tangent space $T_xM^n$ and let $\{e_{n+1},\cdots, e_m\}$ be an orthonormal basis of normal space $T^{\perp}_xM^n$. The mean curvature vector $H$ at $x$ is

The squared mean curvature of the submanifold $M^n$ in $\widetilde{M}^m$ is defined as

Also, we set

Let $K(e_{i}{\wedge}e_{j})$ be the sectional curvature of the plane section spanning $e_{i}$ and $e_{j}$ at $x{\in}M^n$. Subsequently, the scalar curvature $\tau(x)$ of $M^n$ is given by

and the normalized scalar curvature $\rho$ of $M^n$ at $x$ is defined as

The Casorati curvature ${\cal{C}}$ of the submanifold $M^n$ is the squared norm of the second fundamental form $h$ over dimension $n$ and is given by

If $L$ is an $l$-dimensional subspace of $T_xM^n$, where $l{\geqslant}2$ and $\{e_1,{\cdots},e_l\}$ is an orthonormal basis of $L$, the scalar curvature $\tau(L)$ of the $l$-plane section $L$ is defined as

and the Casorati curvature of the subspace $L$, denoted by ${\cal{C}}(L)$, is given by

The generalized normalized $\delta$–Casorati curvatures $\delta_C(r;$$n-1)$ and $\widehat{\delta_C}(r;n-1)$ of the submanifold $M^n$ are defined for a positive real number $r{\ne}n(n-1)$ as

if $0<r<n^2-n$; and

if $r>n^2-n$.

By Gauss equation, we get

where $K(e_{i}{\wedge}e_{j})$ and $\widetilde{K}(e_{i}{\wedge}e_j)$ denote the sectional curvatures of the plane section spanned by $e_{i}$ and $e_{j}$ at $x$ in the submanifold $M^n$ and in the ambient manifold $\widetilde{M}^m$, respectively. By Eqs. $(2)$ and $(25)$, we have

where ${{\parallel}P^T{\parallel}}^2_{N_1^p} = {\sum\nolimits_{i = 1}^{p}} g(P^T,e_i)^2$, ${{\parallel}P^T{\parallel}}^2_{N_2^q} = {\sum\nolimits_{s = p+1}^{n}}g(P^T,e_s)^2$.

Definition 2.1. For the differential function $f$ on $M^n$, the Laplacian ${\Delta}f$ and gradient ${\nabla}f$ of $f$ are defined by

for any vector field $X$ that is tangent to $M^n$.

Lemma 2.1.^[9] Let $M^n = N_1^p{\times}_fN_2^q$ be a warped product submanifold of $\widetilde{M}^m$. The relation between the sectional curvature and Laplacian ${\Delta}f$ of $f$ is

Lemma 2.2.^[10] Let $N_1$ be a Riemannian submanifold of a Riemannian manifold $(N_2,\bar{g})$, $\varphi:N_2{\rightarrow}{\mathbb{R}}$ be a differentiable function and consider the constrained extremum problem

If $x_0{\in}N_1$ is a solution of the problem $(31)$, then

${\rm{(i)}}$ $({\rm{grad}} \ \varphi)(x_0){\in}T_{x_0}^{\perp}N_1$;

${\rm{(ii)}}$ the bilinear form $\Lambda:T_{x_0}N_1{\times}T_{x_0}N_1{\rightarrow}{\mathbb{R}}$ defined by

is positive semi-definite, where $h_1$ is the second fundamental form of $N_1$ in $N_2$ and grad $\varphi$ is the gradient of $\varphi$.

3.   Proof of the theorem

Proof of Theorem 1.1 From Eqs. $(26)$, $(27)$, $(30)$, and the Gauss equation, we obtain

We define the following quadratic polynomial ${\cal{P}}$ in the components of the second fundamental form as

where $L$ denotes the hyperplane of $T_xM^n$. Without loss of generality, we can suppose that $L$ is spanned by $e_1,e_2,\cdots,e_{n-1}$. From Eqs. $(33)$ and $(34)$, we have

We consider the quadratic forms

defined by

Then by Eqs. $(35)$ and $(37)$, we derive

Next, for $\alpha$, we consider the extremum problem

where $K^{\alpha}$ is a real constant(see Ref. [11]). The partial derivatives of function $\varphi_{\alpha}$ are

Applying Lemma $2.2$, for an optimal solution $(h_{11}^{\alpha},h_{22}^{\alpha},{\cdots},h_{nn}^{\alpha})$ of the minimum problem, vector $grad \ \varphi_{\alpha}$ is normal at $\varGamma$ and collinear with the vector $(1,1,\cdots,1)$.

From Eq. $(40)$ and Lemma $2.2$, we derive that a critical point of the problem has the following form:

We fixed an arbitrary point $x{\in}\varGamma$. According to Lemma $2.2$, we deduce that the corresponding bilinear form $\Lambda:T_x{\varGamma}{\times}T_x{\varGamma}{\rightarrow}{\mathbb{R}}$ is given by

where $h'$ is the second fundamental form of $\varGamma$ in ${\mathbb{R}}^n$ and $g$ is the inner product on ${\mathbb{R}}^n$.

By Eq. $(40)$, for $i$, $j\in\{1,\cdots,p\}$, $i{\neq}j$ and $s$, $t\in\{p+1,\cdots,n\}$, $s{\neq}t$, we get

Note that

where

Since $g$ is the inner product on ${\mathbb{R}}^n$, $g_{ij}$ is constant, and $\varGamma_{ij}^k$ is $0$. Then, we have

The Hessian matrix of $\varphi_{\alpha}$ is

As $\varGamma$ is totally geodesic in ${\mathbb{R}}^n$, we consider a vector $X = (X_1,$$X_2,\cdots, X_n)$ tangent to $\varGamma$ at an arbitrary point $x$ on $\varGamma$, that is, we verify the relation $\sum\nolimits_{i = 1}^nX_i = 0$(see Ref. [11]). Next, we prove $\Lambda(X,$$X)\geqslant 0$.

${\rm{(i)}}$ For $n = 1$, we have two possibilities

Since $X_1 = 0$, in this two cases

${\rm{(ii)}}$ For $n = 2$. we have three possibilities

For ${\rm{(a)}}$,

For ${\rm{(b)}}$, ${\rm{Hess}}_{\varphi_{\alpha}}$ is positive definite, i.e., ${\rm{Hess}}_{\varphi_{\alpha}}(X,X) > 0$. For ${\rm{(c)}}$, ${\rm{Hess}}_{\varphi_{\alpha}}$ is positive semi-definite, i.e., ${\rm{Hess}}_{\varphi_{\alpha}}(X,X)\geqslant0$.

${\rm{(iii)}}$ For $n\geqslant3$, when $n = p$, we have

When $n>p$, let

Note that

Thus, all eigenvalues of $A$ are greater than $0$, i.e., $A$ is positive definite.

Since $n>p$, when $n-p = q = 1$, we have

$B$ is positive definite. When $n-p = q\geq2$, we have

Since

we have

that is,

We prove that all eigenvalues of $B$ are greater than or equal to $0$, i.e., $B$ is positive semi-definite. Thus, we prove that ${\rm{Hess}}_{\varphi_{\alpha}} (X,X)\geqslant0$.

Combining ${\rm{(i)}}$, ${\rm{(ii)}}$ and ${\rm{(iii)}}$, we have

Hence, by Eq. $(41)$, the point $(h_{11}^{\alpha},h_{22}^{\alpha},\cdots,h_{nn}^{\alpha})$ is a global minimum point. From Eqs. $(37)$ and $(41)$, we have

We divide the proof of Theorem $1.1$ into two main cases, according to $0<r<n(n-1)$ or $r>n(n-1)$.

Case 1: $0<r<n(n-1)$. In this case, using $(38)$ and $(61)$, we derive

By Eqs. $(34)$ and $(62)$, we derive

Taking the infimum over all tangent hyperplanes $L$ of $T_xM^n$ in $(63)$, we obtain

Case 2: $r>n(n-1)$. Similarly, using the same method, we obtain

Equalities hold in (64) and (65) at a point $x{\in}M^n$ if and only if inequalities $(35)$ and $(61)$ become equalities. Thus, we have

By choosing an orthonormal basis such that $e_{n+1}$ is in the direction of the mean curvature vector, we have

where $f_1 = \dfrac{K^{n+1}}{(n^2-n+qr-r)^2+p^2r^2}$ is a function on $M^n$.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (12026262).

Conflict of interest

The authors declare that they have no conflict of interest.