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# Characterizing small spheres in a unit sphere by Fischer–Marsden equation

*Journal of Inequalities and Applications*
**volume 2022**, Article number: 118 (2022)

## Abstract

We use a nontrivial concircular vector field **u** on the unit sphere \(\mathbf{S}^{n+1}\) in studying geometry of its hypersurfaces. An orientable hypersurface *M* of the unit sphere \(\mathbf{S}^{n+1}\) naturally inherits a vector field **v** and a smooth function *ρ*. We use the condition that the vector field **v** is an eigenvector of the de-Rham Laplace operator together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field **v** to find a characterization of small spheres in the unit sphere \(\mathbf{S}^{n+1}\). We also use the condition that the function *ρ* is a nontrivial solution of the Fischer–Marsden equation together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field **v** to find another characterization of small spheres in the unit sphere \(\mathbf{S}^{n+1}\).

## 1 Introduction

The study of the geometry of hypersurfaces in a sphere is a captivating subject in differential geometry that has been investigated by many researchers (see, e.g., [4, 7, 8, 11, 12, 20–23, 26, 31, 32, 35]), one of the most interesting problems in this field, still unsolved, being the famous *Chern Conjecture for isoparametric hypersurfaces* (see [39, Problem 105] and also the remarkable review paper [28]). We would like to emphasize that several notable results have been established in this field over time. For instance, Okumura [24] provided a criterion for a hypersurface of constant mean curvature in an odd-dimensional sphere to be totally umbilical. Later, do Carmo and Warner [13], as well as Wang and Xia [34], investigated the rigidity of hypersurfaces in spheres, while Chen characterized minimal hypersurfaces in the same ambient space [6]. Some global pinching results concerning minimal hypersurfaces in spheres were obtained by Shen [30]. Other interesting pinching theorems were derived in [1, 18, 36–38]. Recent results on the geometry of hypersurfaces in spheres were obtained in [2, 3, 27, 29, 40].

One of the interesting but challenging problems in submanifold geometry is characterizing small spheres (non-totally geodesic totally umbilical spheres) in a unit sphere \(\mathbf{S}^{n+1}\) (see [19]). On a Riemannian manifold \((M,g)\), the Ricci operator *T* is defined using Ricci tensor *S*, namely \(S(X,Y)=g(TX,Y)\), \(X\in \mathfrak{X}(M)\), where \(\mathfrak{X}(M)\) is the Lie algebra of smooth vector fields on *M*. Similarly, the rough Laplace operator on the Riemannian manifold \((M,g)\), \(\Delta : \mathfrak{X}(M)\rightarrow \mathfrak{X}(M)\) is defined by

where ∇ is the Riemannian connection and \(\{ e_{1},\ldots,e_{m} \} \) is a local orthonormal frame on *M*, \(m=\dim M\). The rough Laplace operator is used in finding characterizations of spheres as well as of Euclidean spaces (cf. [15, 17]). Recall that the de-Rham Laplace operator \(\square : \mathfrak{X}(M)\rightarrow \mathfrak{X}(M)\) on a Riemannian manifold \((M,g)\) is defined by (cf. [14], p.83)

and is used to characterize a Killing vector field on a compact Riemannian manifold. It is known that if *ξ* is a Killing vector field on a Riemannian manifold \((M,g)\) or soliton vector field of a Ricci soliton \((M,g,\xi ,\lambda )\), then \(\square \xi =0\) (cf. [10]). Also, Fischer and Marsden considered in [16] the following differential equation on a Riemannian manifold \((M,g)\):

where \(Hess(f)\) is the Hessian of a smooth function *f* and Δ is the Laplace operator acting on smooth functions of *M*. They conjectured that if a compact Riemannian manifold admits a nontrivial solution of the differential equation (2), then it must be an Einstein manifold. Recent investigations on manifolds satisfying the Fischer–Marsden equation were done in [5, 9, 25, 33].

Consider the sphere \(\mathbf{S}^{n+1}\) as hypersurface of the Euclidean space \(\mathbf{R}^{n+2}\) with unit normal *ξ* and shape operator \(B=-\sqrt{c}I\), where *I* denotes the identity operator. For the constant vector field \(\overrightarrow{a}=\frac{\partial }{\partial x^{1}}\) on the Euclidean space \(\mathbf{R}^{n+2}\), where \(x^{1},\ldots,x^{n+2}\) are Euclidean coordinates on \(\mathbf{R}^{n+2}\), we denote by **u** the tangential projection of \(\overrightarrow{a}\) on the unit sphere \(\mathbf{S}^{n+1}\). Then we have

where \(\overline{f}= \langle \overrightarrow{a},\xi \rangle \), \(\langle , \rangle \) is the Euclidean metric on \(\mathbf{R}^{n+2}\). Taking covariant derivative in the above equation with respect to a vector field *X* on the unit sphere \(\mathbf{S}^{n+1}\) and using Gauss–Weingarten formulae for hypersurface, we conclude

where ∇̅ is the Riemannian connection on the unit sphere \(\mathbf{S}^{n+1}\) with respect to the canonical metric *g* and grad*f̅* is the gradient of the smooth function *f̅* on \(\mathbf{S}^{n+1}\). Thus, **u** is a concircular vector field on the unit sphere \(\mathbf{S}^{n+1}\). Now consider the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) defined by

Then it follows that \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) is a hypersurface of the unit sphere \(\mathbf{S}^{n+1}\) with unit normal vector field *N* given by

We denote by the same letter *g* the induced metric on the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) and denote by ∇ the Riemannian connection with respect to the induced metric *g*. Then, by a straightforward computation, we find that

Thus, the shape operator *A* of the hypersurface \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) is given by

where *α* is the mean curvature of the hypersurface \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \). It is clear that *α* is a nonzero constant as \(0< c<1\). Now, denote by **v** the tangential projection of the vector field **u** to the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) and define \(\rho =g ( \mathbf{u},N )\). Then we have

However, we can easily see using the definitions of **u** and *N* that

where *f* is the restriction of *f̅* to \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \). Thus, \(\rho =-\alpha f\). Taking covariant derivative in equation (6) and using Gauss–Weingarten formulae for hypersurface, we conclude on using equations (3) and (5) by equating tangential components that

for \(X\in \mathfrak{X} ( \mathbf{S}^{n} ( \frac{1}{c^{2}} ) )\). Also, we have \(\operatorname{grad}f=\mathbf{v}\). Thus, the rough Laplace operator Δ acting on **v** and the Laplace operator acting on the smooth function *ρ* are respectively given by

The Ricci operator *T* of the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) is given by

Thus, we observe that the vector field **v** on the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) satisfies

Also, using equation (8), we see that the Hessian of *ρ* is given by

for \(X,Y\in \mathfrak{X} (\mathbf{S}^{n} ( \frac{1}{c^{2}} ) )\), and using the above equation with expression for Ricci tensor and equation (8), we see that the function *ρ* on the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) satisfies the Fischer–Marsden equation

Thus, in view of equations (9) and (10), the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) admits a vector field **v** that is an eigenvector of the de-Rham Laplace operator with eigenvalue \((n-2)(1+\alpha ^{2})\), and it admits a smooth function *ρ* that is a solution of the Fischer–Marsden differential equation. These raise two questions: (i) Given a compact hypersurface *M* of the unit sphere \(\mathbf{S}^{n+1}\) that admits a vector field **v**, which is the eigenvector of de-Rham Laplace operator □ corresponding to positive eigenvalue, is this hypersurface necessarily isometric to a small sphere? (ii) Given a compact hypersurface *M* admitting a vector field **v** and a smooth function *ρ* with gradient \(\operatorname{grad}\rho =-A\mathbf{v}\) a nontrivial solution of the Fischer–Marsden differential equation, is this hypersurface necessarily isometric to a small sphere? In this paper, we answer these questions (cf. Theorem 3.1 and Theorem 3.2).

## 2 Preliminaries

Let *M* be an orientable hypersurface of the unit sphere \(\mathbf{S}^{n+1}\) with unit normal vector field *N* and shape operator *A*. We denote the canonical metric on \(\mathbf{S}^{n+1}\) by *g* and denote by the same letter *g* the induced metric on the hypersurface *M*. Let ∇̅ and ∇ be the Riemannian connections on the unit sphere \(\mathbf{S}^{n+1}\) and on the hypersurface *M*, respectively. Then we have the following fundamental equations of the hypersurface:

The curvature tensor field *R*, the Ricci tensor *S*, and the scalar curvature *τ* of the hypersurface *M* are given by

and

where \(X,Y,Z\in \mathfrak{X}(M)\) and \(\alpha =\frac{1}{n}\operatorname{Tr} A\) is the mean curvature of the hypersurface *M* and \(\Vert A \Vert ^{2}=\operatorname{Tr} A^{2}\). The Codazzi equation of hypersurface gives

where

Taking a local orthonormal frame \(\{ e_{1},\ldots,e_{n} \} \) on the hypersurface *M*, equation (15) yields

Let **u** be the concircular vector field on the unit sphere \(\mathbf{S}^{n+1}\) considered in the previous section, which satisfies equation (3), where *f̅* is the function defined on \(\mathbf{S}^{n+1}\) by \(\overline{f}= \langle \overrightarrow{a},\xi \rangle \). We denote the restriction of *f̅* to the hypersurface *M* by *f* and the tangential projection of the vector field **u** on *M* by **v**. Then we have

We call the vector field **v** the induced vector field on the hypersurface *M*. We also call the functions *ρ* and *f* the support function and the associated function, respectively, of the hypersurface *M*. Note that grad*f* is the tangential component of grad*f̅*, i.e.,

while the normal component of grad*f* is

that is, on using equations (3) and (17), we have

Taking covariant derivative in equation (17) and using equations (3) and (11), we get on equating tangential and normal components

### Lemma 2.1

*Let* *M* *be a compact hypersurface of the unit sphere* \(\mathbf{S}^{n+1}\) *with induced vector field* **v**, *support function* *ρ*, *and associated function* *f*. *Then*

### Proof

Using equation (19), we have

and using equation (18), we get

Integrating the above equation, we get the result. □

### Lemma 2.2

*Let* *M* *be a compact hypersurface of the unit sphere* \(\mathbf{S}^{n+1}\) *with induced vector field* **v**, *support function* *ρ*, *and associated function* *f*. *Then*

### Proof

Note that we have

Integrating this equation and using the second equation in (19), we get the result. □

## 3 Characterizations of small spheres

Let **u** be the concircular vector field on the unit sphere \(\mathbf{S}^{n+1}\) and *M* be its orientable non-totally geodesic hypersurface with mean curvature *α* and induced vector field **v**, potential function *ρ*, and associated function *f*. In this section we find different characterizations of the small spheres in \(\mathbf{S}^{n+1}\).

### Theorem 3.1

*Let* *M* *be an orientable non*-*totally geodesic compact and connected hypersurface of the unit sphere* \(\mathbf{S}^{n+1}\), \(n\geq 2\), *with induced vector field* **v**, *nonzero potential function* *ρ*, *and associated function* *f*. *Then* \(\square \mathbf{v}=\lambda \mathbf{v}\) *for a constant* *λ*, *and the inequality*

*holds if and only if* *α* *is a constant and* *M* *is isometric to the small sphere* \(\mathbf{S}^{m} ( 1+\alpha ^{2} ) \).

### Proof

Suppose that **v** satisfies

where *λ* is a constant. Using equation (13), we have

Now, using equation (18), we get

which gives the rough Laplace operator acting on the vector field **v** as

where we have used equation (16). The above equation in view of equations (18) and (19) becomes

Thus, equations (20), (21), and (22) imply

Taking the inner product in the above equation with **v**, we get

By integrating the above equation and using Lemma 2.2, we conclude

Now, using equation (13) in the above equation, we arrive at

which in view of Lemma 2.1 gives

Therefore, we derive

Note that equation (18) implies

and Bochner’s formula gives

Using equation (18), we have

and

Hence we derive

Thus, from equation (24), we have

that is,

Combining equations (23) and (25) (retaining out of \(2S ( \mathbf{v},\mathbf{v} ) \) one term in (24)), we get

The above equation gives immediately

Using the condition in the statement in the above equation, we get

However, as the support function \(\rho \neq 0\), we get \(\Vert A \Vert ^{2}=n\alpha ^{2}\), and this equality in view of Schwartz’s inequality holds if and only if

Using a local orthonormal frame \(\{ e_{1},\ldots,e_{n} \} \) in the above equation, we get

and combining the above equation with equation (16), we get

As \(n\geq 2\), we conclude that the mean curvature *α* is a constant, and by equation (26) we see that *M* is totally umbilical hypersurface. Hence, by equation (12), we see that *M* is isometric to the small sphere \(\mathbf{S}^{n} ( 1+\alpha ^{2} ) \).

Conversely, if \((M,g)\) is isometric to the sphere \(\mathbf{S}^{m} ( 1+\alpha ^{2} ) \), then choosing positive constant *c* such that

it is clear that \(0< c<1\). We know by equation (9) that potential function *ρ* on the small sphere \(\mathbf{S}^{n} ( \frac{1}{c^{2}} ) \) satisfies

where *λ* is obviously a constant. Also, we have the Ricci curvature

and, in view of Lemma 2.1 and \(\rho =-\alpha f\) for the small sphere, we deduce

Also, on using

we have

Thus, equations (27), (28), and (29) imply that the conditions in the statement of Theorem hold. Finally, observe that if \(\rho =0\) on the small sphere \(\mathbf{S}^{m} ( 1+\alpha ^{2} )\) with constant \(\alpha \neq 0\), we get \(f=0\), and consequently \(\mathbf{v}=0\). Then, by equation (6), we get \(\mathbf{u}=0\), and equation (3) implies \(\overline{f}=0\). Thus, with assumption \(\rho =0\), we reach \(\overrightarrow{a}=0\), hence a contradiction to the fact that \(\overrightarrow{a}\) is a constant unit vector field on the Euclidean space \(\mathbf{R}^{n+2}\). Hence all the requirements in the statement are met. □

Recall that if an *n*-dimensional Riemannian manifold \((M,g)\) admits a nontrivial solution of the Fischer–Marsden differential equation (2), \(n>2 \), then the scalar curvature *τ* is a constant (cf. [16]) and the nontrivial solution *f* satisfies

### Theorem 3.2

*Let* *M* *be an orientable non*-*totally geodesic compact and connected hypersurface of the unit sphere* \(\mathbf{S}^{n+1}\), \(n>2\), *with induced vector field* **v**, *nonzero potential function* *ρ*, *and associated function* *f*. *Then the potential function* *ρ* *is a nontrivial solution of the Fischer–Marsden equation* (2) *and the inequality*

*holds if and only if* *α* *is a constant and* *M* *is isometric to the small sphere* \(\mathbf{S}^{m} ( 1+\alpha ^{2} )\).

### Proof

Let *M* be an orientable non-totally geodesic compact and connected hypersurface of the unit sphere \(\mathbf{S}^{n+1}\), \(n>2\), with induced vector field **v**, nonzero potential function *ρ*, and associated function *f*. Suppose that *ρ* is the nontrivial solution of the Fischer–Marsden equation (2). Then, by equation (30), we have

Using equations (16) and (19), we find

and consequently, equation (19) implies

Using equation (31) with the above equation, we get

Integrating the above equation and using Lemma 2.2, we get

Note that *τ* is a constant and equations (19) and (31) imply

Also, equation (13) gives

which in view of equation (34) and Lemma 2.1 implies

Combining the above equation with equation (33), we arrive at

Now, using

in the above equation, we get

Using now the hypothesis

in equation (35), we conclude

However, as the function \(\rho \neq 0\) on connected *M*, we have \(\Vert A \Vert ^{2}=n\alpha ^{2}\). But, in view of Schwartz’s inequality, this equality holds if and only if \(A=\alpha I\). Hence, *M* being non-totally geodesic hypersurface and \(n>2\), *M* is isometric to the small sphere \(\mathbf{S}^{n}(1+\alpha ^{2})\).

Conversely, as we have seen in the introduction, on the small sphere \(\mathbf{S}^{n}(1+\alpha ^{2})\), the function *ρ* is a solution of Fischer–Marsden equation (cf. equation (10)). Now, the Ricci curvature

together with Lemma 2.1 and \(\rho =-f\alpha \) implies

Also, we have

and we derive

As seen in the proof of Theorem 3.1, we have that the function \(\rho \neq 0\). Thus, by equations (36) and (37), we can see immediately that all the requirements are met in the statement for the small sphere \(\mathbf{S}^{n}(1+\alpha ^{2})\). □

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This work is supported by the Researchers Supporting Project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia.

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Bin Turki, N., Deshmukh, S. & Vîlcu, GE. Characterizing small spheres in a unit sphere by Fischer–Marsden equation.
*J Inequal Appl* **2022**, 118 (2022). https://doi.org/10.1186/s13660-022-02855-4

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DOI: https://doi.org/10.1186/s13660-022-02855-4