Normal curvature of pseudo-umblical submanifolds in a sphere

TL;DR

This paper investigates conditions under which a compact pseudo-umbilical submanifold in a sphere must be totally geodesic, linking normal curvature, scalar curvature, and second fundamental form.

Contribution

It establishes new criteria involving normal curvature and scalar curvature that guarantee a pseudo-umbilical submanifold is totally geodesic.

Findings

Normal curvature, scalar curvature, and second fundamental form conditions imply total geodesicity

Provides criteria for identifying totally geodesic submanifolds in spheres

Enhances understanding of geometric properties of pseudo-umbilical submanifolds

Abstract

Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.

Equations

$$K^{\perp }\le \sigma \text{ , }S>(n-1{)}^2$$

$$⟨h(X,Y),H⟩=\lambda ⟨X,Y⟩$$

$$K^{\perp }=\sum _{i,j,\alpha ,\beta }[R^{\perp }(e_i,e_j,N_{\alpha },N_{\beta }){]}^2$$

$$K^{\perp }\le \sigma \text{ , }S>(n-1{)}^2$$

$$(\nabla h)(X,Y,Z)=(\nabla h)(Y,Z,X)=(\nabla h)(Z,X,Y)$$

$$(\nabla h)(X,Y,Z)$$

$$({\nabla }^{2}h)(X,Y,Z,W)$$

$$-(\nabla h)(Y,{\nabla }_{X}Z,W)-(\nabla h)(Y,Z,{\nabla }_{X}W)$$

$$({\nabla }^{2}h)(X,Y,Z,W)-({\nabla }^{2}h)(Y,X,Z,W)$$

$$-h(Z,R(X,Y)W)$$

$$\sum _{i=1}^{n}h(e_i,e_i)=nH$$

$$A_{h(Y,Z)}X$$

$$R^{\ast }(X)$$

$$S$$

$$g(A_{N}X,Y)=g((h(X,Y),N).$$

$$\sigma$$

$$\Vert A_h{\Vert }^2$$

$$\Vert {\nabla }_h{\Vert }^2$$

$$R(e_k,e_i;e_j,A_{h(e_i,e_j)}{e}_k)$$

$$+g(A_{h(e_i,e_j)}{e}_k,A_{h(e_i,e_j)}{e}_k)-g(A_{h(e_k,e_j)}{e}_i,A_{h(e_i,e_j)}{e}_k)$$

$$={\delta }_{ij}g(h(e_k,e_k),h(e_i,e_j))-{\delta }_{kj}g(h(e_i,e_j),h(e_i,e_k))$$

$$+g(A_{h(e_i,e_j)}{e}_k,A_{h(e_i,e_j)}{e}_k)-g(h(e_k,e_j),h(e_i,A_{h(e_i,e_j)}{e}_k))$$

$$g(h(e_k,e_j),h(e_i,A_{h(e_i,e_j)}{e}_k))=\sum _{i,j,\alpha ,\beta }g(A_{\alpha }{A}_{\beta }{e}_i,e_j)g(A_{\beta }{A}_{\alpha }{e}_i,e_j)$$

$$g(A_{h(e_i,e_j)}{e}_k,A_{h(e_i,e_j)}{e}_k)=\Vert A_h{\Vert }^2$$

$$K^{\perp }$$

$$=\sum _{i,j,\alpha ,\beta }[g(A_{\alpha }{A}_{\beta }{e}_i,e_j)-g(A_{\beta }{A}_{\alpha }{e}_i,e_j){]}^2$$

$$=2\sum _{i,j,\alpha ,\beta }g(A_{\alpha }{e}_i,A_{\beta }{e}_j{)}^2-2\sum _{i,j,\alpha ,\beta }g(A_{\alpha }{A}_{\beta }{e}_i,e_j)g(A_{\beta }{A}_{\alpha }{e}_i,e_j)$$

$$\Rightarrow \frac{1}{2}{K}^{\perp }-$$

$$\sum _{i,j,k}{R}^{\perp }(e_k,e_i;h(e_k,e_j),h(e_i,e_j))$$

$$+\sum _{i,j,k}[R(e_i,e_k,e_j,A_{h(e_i,e_j)}{e}_k)-R(e_k,e_i,e_k,A_{h(e_i,e_j)}{e}_i)]$$

$$-\sum _{i,j,k}g(A_{h(e_k,e_k)}{e}_j,A_{h(e_i,e_j)}{e}_i).$$

$$\sum _{i,j,k}{R}^{\perp }(e_k,e_i;h(e_k,e_j),h(e_i,e_j))$$

$$-g(A_{h(e_i,e_j)}{e}_j,e_i)+g(h(e_j,e_j),h(e_k,e_k))$$

$$-g(A_{h(e_k,e_j)}{e}_k,A_{h(e_i,e_j)}{e}_i)]$$

$$R^{\ast }(e_j)=(n-1)e_j-\sum _k^n{A}_{h(e_j,e_k)}{e}_k+\sum _k^n{A}_{h(e_k,e_k)}{e}_j$$

$$\sum _{i,j}g(R^{\ast }(e_j),A_{h(e_i,e_j)}$$

$$+\sum _{i,j,k}g(A_{h(e_k,e_k)}{e}_j,A_{h(e_i,e_j)}{e}_i)-\sum _{i,j,k}g(A_{h(e_j,e_k)}{e}_k,A_{h(e_i,e_j)}{e}_i)$$

$$\sum _{i,j,k}g(A_{h(e_j,e_k)}{e}_k,A_{h(e_i,e_j)}{e}_i)$$

$$+\sum _{i,j,k}g(A_{h(e_k,e_k)}{e}_j,A_{h(e_i,e_j)}{e}_i)-\sum _{i,j}g(R^{\ast }(e_j),A_{h(e_i,e_j)}{e}_i)$$

Full text

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then $M^{n}$ is totally geodesic.

1. Introduction

Let $S^{n+p}$ be an $(n+p)$ dimensional unit sphere and $M$ be a compact $n$-dimensional submanifold isometrically immersed in $S^{n+p}$. Let $h$ be the second fundamental form of the immersion and $H$ be the mean curvature vector. Denote by $\langle\cdot,\cdot\rangle$ the scalar product of $S^{n+p}$. If there exists a function $\lambda$ on $M$ such that

[TABLE]

for any tangent vectors $X$, $Y$ on $M$, then $M$ is called a Pseudo-umbilical submanifold of $S^{n+p}$ ([1],[2]). It is clear that $\lambda\geq 0$. If the mean curvature vector $H=0$ identically, then $M$ is called a minimal submanifold of $S^{n+p}$. Every minimal submanifold of $S^{n+p}$ is itself a pseudo-umbilical submanifold.

Let $M$ be a compact $n$-dimensional Pseudo-umbilical submanifold of the unit sphere $S^{n+p}$ with normal bundle $\nu$. We denote by $R^{\bot}$ the curvature tensor field corresponding to the normal connection $\nabla^{\perp}$ in the normal bundle $\nu$ of $M^{n}$, and define $K^{\bot}:M\rightarrow R$ by

[TABLE]

where $\{e_{1},...,e_{n}\}$ is a local orthonormal frame on $M^{n}$ and $\{N_{1},...,N_{p}\}$ is a local field of orthonormal normals. we call the function $K^{\bot}$ the normal curvature of $M^{n}$.

There are several results for compact minimal submanifolds in a unit sphere. Simon [6], in his famous paper gave a pinching theorem, which led to an intrinsic rigidity result. Later on, Simon’s work was improved by Sakaki [4] for arbitrary codimension. Shen [5] further improved the result of Sakaki but only for dimension $n=3$. Deshmukh [3] partially generalized the result of Shen and proved $M^{n}$ to be totally geodesic with the impositions of certain conditions on normal curvature $K^{\bot}$, scalar curvature $S$ and square of the length of second fundamental form $\sigma$. On the other hand, Choudhary [2] studied pseudo-umbilical hypersurfaces in the unit sphere. Inspired by all the above developments, we study Pseudo-umbilical submanifolds in a unit sphere and prove the following result.

Theorem 1.1**.**

Let $M$ be compact pseudo-umbilical submanifold of the unit sphere $S^{n+p}$ with normal bundle $\nu$. If normal curvature $K^{\bot}$, scalar curvature $S$ and square of the length of second fundamental form $\sigma$ satisfy

[TABLE]

then $M^{n}$ is totally geodesic.

Infact, we have worked to investigate the normal curvature of Pseudo-umbilical submanifolds in a sphere and tried to generalize the results due to Deshmukh [3].

2. Preliminaries

Let $M$ be an n-dimensional pseudo-umbilical submanifold of the unit sphere $S^{n+p}$ with normal bundle $\nu$. We denote by $g$ the Riemannian metric on $S^{n+p}$ as well as the induced metric on $M$. Let $\nabla$ be the Riemannian connection and $A$ be the shape operator of the submanifold $M$. Then the second fundamental form $h$ of $M^{n}$ satisfies

[TABLE]

for $X,Y,Z\in\mathcal{X}(M)$, where $\mathcal{X}(M)$ is the Lie algebra of smooth vector fields on $M$ and $(\nabla h)(X,Y,Z)$ is defined by

[TABLE]

where $\nabla^{\perp}$ is the connection defined in $\nu$. The second covariant derivative $(\nabla^{2}h)(X,Y,Z,W)$ of the second fundamental form is given by

[TABLE]

for $X,Y,Z,W\in\mathcal{X}(M)$. The Ricci identity is given by

[TABLE]

for $X,Y,Z,W\in\mathcal{X}(M)$, where $R^{\perp}$ and $R$ are the curvature tensors of the connections $\nabla^{\perp}$ and $\nabla$ respectively. Since $M^{n}$ is a Pseudo-umbilical submanifold, then for a local orthonormal frame $\{e_{1},...,e_{n}\}$ of $M^{n}$ we have

[TABLE]

We define the symmetric operator $R^{*}$ by using the Ricci tensor Ric in the following way $Ric(X,Y)=g(R^{*}(X),Y)$, for $X,Y\in\mathcal{X}(M)$. Then the Gauss equation gives

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where $A_{N},N\in\nu$ is the Weingarten map with respect to the normal $N$, satisfying

[TABLE]

We define

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Now, we state the following lemmas which are required for the proof of our main theorem.

Lemma 2.1**.**

Let $M^{n}$ be an immersed pseudo-umbilical submanifold of the unit sphere $S^{n+p}$, then for a local orthonormal frame $\{e_{1},\ldots,e_{n}\}$, we have $\sum\limits_{i,j,k}R(e_{k},e_{i};e_{j},A_{h(e_{i},e_{j})}e_{k})=n^{2}H^{2}-\sigma+{\|A_{h}\|}^{2}+\frac{1}{2}K^{\perp}-\sum\limits_{i,j,\alpha,\beta}g(A_{\alpha}e_{i},A_{\beta}e_{j})^{2}$, where $A_{\alpha}\equiv A_{N_{\alpha}}$ and $\{N_{1},\ldots,N_{p}\}$ is a local field of orthonormal normals.

Proof.

Using the Gauss equation, we have

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Using the fact that, $A_{h(e_{i},e_{j})}e_{k}=\sum\limits_{\alpha}g(A_{\alpha}e_{i},e_{j})A_{\alpha}e_{k}$ , we obtain

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We also have

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On the other hand, using the Ricci equation $R^{\perp}(X,Y;N_{1},N_{2})=g([A_{N_{1}},A_{N_{2}}](X),Y)$, for $X,Y\in\mathcal{X}(M),N_{1},N_{2}\in\nu,$ we have

[TABLE]

where we have used $\sum\limits_{i,j,\alpha,\beta}g(A_{\alpha}e_{j},A_{\beta}e_{i})^{2}=\sum\limits_{i,j,\alpha,\beta}g(A_{\beta}e_{j},A_{\alpha}e_{i})^{2}$ which follows from the symmetry of $A_{\alpha}$ and $A_{\beta}$.

Finally, in the light of equations (2.9), (2.10) and (2.11) and using the fact that $M^{n}$ is a Pseudo-umbilical submanifold, equation (2.8) reduces to

$\sum\limits_{i,j,k}R(e_{k},e_{i};e_{j},A_{h(e_{i},e_{j})}e_{k})=n^{2}H^{2}-\sigma+{\|A_{h}\|}^{2}+\frac{1}{2}K^{\perp}-\sum\limits_{i,j,\alpha,\beta}g(A_{\alpha}e_{i},A_{\beta}e_{j})^{2}$

and this completes the proof of the lemma. ∎

Next, we prove

Lemma 2.2**.**

Let $M^{n}$ be a pseudo-umbilical submanifold of the unit sphere $S^{n+p}$, then for a local orthonormal frame $\{e_{1},\ldots,e_{n}\}$, we have

[TABLE]

Proof.

Let $M^{n}$ be a pseudo-umbilical submanifold of the unit sphere $S^{n+p}$. Then, using (2.4) in the Ricci equation, and in the light of (2.7), we obtain

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Now, taking into consideration equation (2.5), we have

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Taking inner product with $A_{h(e_{i},e_{j})}e_{i}$ in the above equation, we obtain

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So, in the light of equation (2.7), above equation gives

[TABLE]

where, we have used the relationship between operator $R^{*}$ and Ricci tensor Ric. Thus, using using (2.13) in (2.14), we have the required result. ∎

3. Main Results

Proof of Theorem 1.1.

Let $M^{n}$ be a compact Pseudo-umbilical submanifold of $S^{n+p}$ satisfying the hypothesis of the theorem. We define a function

$F:M\rightarrow R\hskip 10.00002pt\text{by}\hskip 10.00002ptF=\frac{1}{2}\sigma.$

Then, the Laplacian $\triangle F$ of the function $F$ can be computed as

[TABLE]

In view of equation (2.1) and the Ricci identity (2.2), above equation gives

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where, we have assumed $\mu=\sum\limits_{i,j,k}g((\nabla^{2}h)(e_{j},e_{i},e_{k},e_{k}),h(e_{i},e_{j}))$. Now, taking into account equation (2.7), one can write above equation as follows

[TABLE]

On the other hand,

[TABLE]

since $A_{h(e_{i},e_{j})}e_{k}=\sum\limits_{\alpha}g(A_{\alpha}e_{i},e_{j})A_{\alpha}e_{k}$, we obtain

[TABLE]

Using (3.2) in (3.1) and taking view of lemma 2.1 and lemma 2.2, we have

[TABLE]

In the light of (2.7), we also have

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Let us suppose that $\sigma=\sum\limits_{\alpha}\|A_{\alpha}\|^{2}$. Then, from (3.3) and (3.4), we have

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Integrating over $M^{n}$, we obtain

[TABLE]

Now, as per the hypothesis of the theorem $S>(n-1)^{2}$, it follows that $n(n-1)+n^{2}H^{2}-\sum\limits_{\alpha}\|A_{\alpha}\|^{2}>(n-1)^{2}$. That is, $n^{2}H^{2}-\sum\limits_{\alpha}\|A_{\alpha}\|^{2}<(n-1)$, and hence, $n^{2}H^{2}-\|A_{\alpha}\|^{2}>(n-1)$, and that $K^{\perp}\leq\sigma$. Thus, in order for (3.5) to hold, we must have $\|A_{\alpha}\|=0$, that is $M^{n}$ is totally geodesic and this proves the theorem. ∎