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

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

$$ \Delta X=\sum_{i=1}^{m} ( \nabla _{e_{i}}\nabla _{e_{i}}X- \nabla _{\nabla _{e_{i}}e_{i}}X ) ,\quad X\in \mathfrak{X}(M), $$

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)

$$ \square =T+\Delta $$

(1)

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 $(M,g)$:

$$ ( \Delta f ) g+fS=Hess(f), $$

(2)

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

$$ \overrightarrow{a}=\mathbf{u}+\overline{f}\xi , $$

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

$$ \overline{\nabla }_{X}\mathbf{u}=-\overline{f}X,\qquad \operatorname{grad}\overline{f}=\mathbf{u}, $$

(3)

where ∇̅ is the Riemannian connection on the unit sphere $\mathbf{S}^{n+1}$ with respect to the canonical metric g and gradf̅ 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

$$ \mathbf{S}^{n} \biggl( \frac{1}{c^{2}} \biggr) = \Biggl\{ \bigl(x^{1},\ldots x^{n+2}\bigr): \sum _{i=1}^{n+1} \bigl( x^{i} \bigr) ^{2}=c^{2}, x^{n+2}=\sqrt{1-c^{2}}, 0< c< 1 \Biggr\} . $$

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

$$ N= \biggl( -\frac{\sqrt{1-c^{2}}}{c}x^{1},\ldots,- \frac{\sqrt{1-c^{2}}}{c}x^{n+1},c \biggr) . $$

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

$$ \overline{\nabla }_{X}N=-\frac{\sqrt{1-c^{2}}}{c}X,\quad X\in \mathfrak{X} \biggl( \mathbf{S}^{n} \biggl( \frac{1}{c^{2}} \biggr) \biggr) . $$

(4)

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

$$ A=\frac{\sqrt{1-c^{2}}}{c}I=\alpha I, $$

(5)

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

$$ \mathbf{u}=\mathbf{v}+\rho N. $$

(6)

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

$$ g ( \mathbf{u},N ) =-\frac{\sqrt{1-c^{2}}}{c}f, $$

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

$$ \nabla _{X}\mathbf{v}=-\bigl(1+\alpha ^{2}\bigr)fX ,\qquad \operatorname{grad}\rho =-\alpha \mathbf{v}, $$

(7)

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

$$ \Delta \mathbf{v}=-\bigl(1+\alpha ^{2}\bigr)\mathbf{v},\qquad \Delta \rho =-n\bigl(1+ \alpha ^{2}\bigr)\rho . $$

(8)

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

$$ TX=(n-1) \bigl(1+\alpha ^{2}\bigr)X. $$

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

$$ \square \mathbf{v}=(n-2) \bigl(1+\alpha ^{2}\bigr)\mathbf{v}. $$

(9)

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

$$\begin{aligned} Hess(\rho ) (X,Y) =&g ( \nabla _{X}\operatorname{grad}\rho ,Y ) \\ =&\alpha \bigl(1+\alpha ^{2}\bigr)fg ( X,Y ) \\ =&-\bigl(1+\alpha ^{2}\bigr)\rho g ( X,Y ) \end{aligned}$$

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

$$ ( \Delta \rho ) g+\rho S=Hess(\rho ). $$

(10)

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:

$$ \overline{\nabla }_{X}Y=\nabla _{X}Y+g ( AX,Y ) N, \qquad \overline{\nabla }_{X}N=-AX,\quad X,Y\in \mathfrak{X}(M) . $$

(11)

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

$$\begin{aligned}& R(X,Y)Z=g(Y,Z)X-g(X,Z)Y+g(AY,Z)AX-g(AX,Z)AY, \end{aligned}$$

(12)

$$\begin{aligned}& S(X,Y)=(n-1)g(X,Y)+n\alpha g(AX,Y)-g(AX,AY), \end{aligned}$$

(13)

and

$$ \tau =n(n-1)+n^{2}\alpha ^{2}- \Vert A \Vert ^{2}, $$

(14)

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

$$ ( \nabla A ) (X,Y)= ( \nabla A ) (Y,X) ,\quad X,Y\in \mathfrak{X}(M), $$

(15)

where

$$ ( \nabla A ) (X,Y)=\nabla _{X}AY-A ( \nabla _{X}Y ). $$

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

$$ n \operatorname{grad}\alpha =\sum_{i=1}^{n} ( \nabla A ) (e_{i},e_{i}) . $$

(16)

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

$$ \mathbf{u}=\mathbf{v}+\rho N,\qquad \rho =g ( \mathbf{u},N ) . $$

(17)

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 gradf is the tangential component of gradf̅, i.e.,

$$ \operatorname{grad}f= [ \operatorname{grad}\overline{f} ] ^{T}, $$

while the normal component of gradf is

$$\begin{aligned}{} [ \operatorname{grad}\overline{f} ] ^{\perp } =&g ( \operatorname{grad} \overline{f},N ) N \\ =&g ( \mathbf{u},N ) N \\ =&\rho N, \end{aligned}$$

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

$$ \operatorname{grad}f=\mathbf{v}. $$

(18)

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

$$ \nabla _{X}\mathbf{v}=-fX+\rho AX,\qquad \operatorname{grad}\rho =-A \mathbf{v},\quad X\in \mathfrak{X}(M). $$

(19)

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

$$ \int _{M} \Vert \mathbf{v} \Vert ^{2}=n \int _{M} \bigl( f^{2}-f\rho \alpha \bigr) . $$

Proof

Using equation (19), we have

$$ \operatorname{div}\mathbf{v}=n(-f+\rho \alpha ),$$

and using equation (18), we get

$$ \operatorname{div} ( f\mathbf{v} ) = \Vert \mathbf{v} \Vert ^{2}+nf(-f+\rho \alpha ). $$

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

$$ \int _{M}\rho \mathbf{v} ( \alpha ) = \int _{M} \bigl[ \alpha g ( A\mathbf{v},\mathbf{v} ) +nf \rho \alpha -n\rho ^{2}\alpha ^{2} \bigr] . $$

Proof

Note that we have

$$\begin{aligned} \operatorname{div} \bigl( \alpha ( \rho \mathbf{v} ) \bigr) =& \rho \mathbf{v} ( \alpha ) +\alpha \operatorname{div} ( \rho \mathbf{v} ) \\ =&\rho \mathbf{v} ( \alpha ) +\alpha \bigl[ \mathbf{v}( \rho )+n\rho (-f+\rho \alpha \bigr] . \end{aligned}$$

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

$$ \int _{M}S(\mathbf{v},\mathbf{v})\leq n \int _{M} ( f-\rho \alpha ) \bigl[(\lambda +1)f-\rho \alpha \bigr] $$

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

$$ \square \mathbf{v}=\lambda \mathbf{v}, $$

(20)

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

$$ T(\mathbf{v})=(n-1)\mathbf{v}+n\alpha A\mathbf{v}-A^{2}\mathbf{v} . $$

(21)

Now, using equation (18), we get

$$ \nabla _{X}\nabla _{X}\mathbf{v}-\nabla _{\nabla _{X}X} \mathbf{v}=-X(f)X+X(\rho )AX+\rho ( \nabla A ) (X,Z), $$

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

$$ \Delta \mathbf{v}=-\operatorname{grad}f+A (\operatorname{grad}\rho ) +n \rho \operatorname{grad}\alpha , $$

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

$$ \Delta \mathbf{v}=-\mathbf{v}-A^{2}\mathbf{v}+n\rho \operatorname{grad}\alpha . $$

(22)

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

$$ (n-2-\lambda )\mathbf{v}-2A^{2}\mathbf{v}+n\alpha A\mathbf{v}+n\rho \operatorname{grad}\alpha =0. $$

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

$$ (n-2-\lambda ) \Vert \mathbf{v} \Vert ^{2}-2 \Vert A \mathbf{v} \Vert ^{2}+n\alpha g ( A\mathbf{v},\mathbf{v} ) +n \rho \mathbf{v} ( \alpha ) =0. $$

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

$$ \int _{M} \bigl[ (n-2-\lambda ) \Vert \mathbf{v} \Vert ^{2}-2 \Vert A\mathbf{v} \Vert ^{2}+2n\alpha g ( A \mathbf{v},\mathbf{v} ) +n^{2}f\rho \alpha -n^{2} \rho ^{2}\alpha ^{2} \bigr] =0. $$

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

$$ \int _{M} \bigl[ -(n+\lambda ) \Vert \mathbf{v} \Vert ^{2}+2S ( \mathbf{v},\mathbf{v} ) +n^{2}f\rho \alpha -n^{2}\rho ^{2}\alpha ^{2} \bigr] =0, $$

which in view of Lemma 2.1 gives

$$ \int _{M}S(\mathbf{v},\mathbf{v})= \int _{M} \bigl[ n^{2} ( -f+\rho \alpha ) ^{2}-nf^{2}- \rho ^{2} \Vert A \Vert ^{2}+2nf\rho \alpha \bigr]. $$

Therefore, we derive

$$ \int _{M} \bigl[ -n(n+\lambda )f^{2}+n(2n+\lambda )f \rho \alpha -n^{2}\rho ^{2}\alpha ^{2}+2S ( \mathbf{v},\mathbf{v} ) \bigr] =0. $$

(23)

Note that equation (18) implies

$$ S ( \mathbf{v},\mathbf{v} ) =S ( \operatorname{grad}f,\operatorname{grad}f ) $$

and Bochner’s formula gives

$$ \int _{M}S(\mathbf{v},\mathbf{v})= \int _{M} \bigl[ ( \Delta f ) ^{2}-Hess(f)^{2} \bigr]. $$

(24)

Using equation (18), we have

$$ \Delta f=n ( -f+\rho \alpha ) $$

and

$$\begin{aligned} Hess(f) ( X,Y ) =&g ( \nabla _{X}\operatorname{grad}f,Y ) \\ =&-fg ( X,Y ) +\rho g(AX,Y). \end{aligned}$$

Hence we derive

$$ Hess(f)^{2}=nf^{2}+\rho ^{2} \Vert A \Vert ^{2}-2nf\rho \alpha . $$

Thus, from equation (24), we have

$$ \int _{M}S(\mathbf{v},\mathbf{v})= \int _{M} \bigl[ n^{2}(-f+ \rho \alpha )^{2}-nf^{2}-\rho ^{2} \Vert A \Vert ^{2}+2nf \rho \alpha \bigr], $$

that is,

$$ \int _{M}S(\mathbf{v},\mathbf{v})= \int _{M} \bigl[ n(n-1)f^{2}+n^{2} \rho ^{2}\alpha ^{2}-\rho ^{2} \Vert A \Vert ^{2}-2n(n-1)f \rho \alpha \bigr]. $$

(25)

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

$$ \int _{M}\rho ^{2} \bigl( \Vert A \Vert ^{2}-n \alpha ^{2} \bigr) = \int _{M} \bigl[ -n \bigl[ (\lambda +1)f^{2}-( \lambda +2)f\rho \alpha +\rho ^{2}\alpha ^{2} \bigr] +S( \mathbf{v}, \mathbf{v}) \bigr] . $$

The above equation gives immediately

$$ \int _{M}\rho ^{2} \bigl( \Vert A \Vert ^{2}-n \alpha ^{2} \bigr) = \int _{M} \bigl[ S(\mathbf{v},\mathbf{v})-n ( f-\rho \alpha ) \bigl( (\lambda +1 ) f-\rho \alpha \bigr) \bigr]. $$

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

$$ \rho ^{2} \bigl( \Vert A \Vert ^{2}-n\alpha ^{2} \bigr) =0. $$

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

$$ A=\alpha I. $$

(26)

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

$$ \sum_{i=1}^{n} ( \nabla A ) (e_{i},e_{i})=\operatorname{grad} \alpha , $$

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

$$ (n-1)\operatorname{grad}\alpha =0. $$

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

$$ c^{2}=\frac{1}{1+\alpha ^{2}}, $$

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

$$ \square \mathbf{v}=\lambda \mathbf{v},\qquad \lambda =(n-2) \bigl( 1+\alpha ^{2} \bigr) , $$

(27)

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

$$ S ( \mathbf{v},\mathbf{v} ) =(n-1) \bigl(1+\alpha ^{2} \bigr) \Vert \mathbf{v} \Vert ^{2}, $$

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

$$ \int _{M}S(\mathbf{v},\mathbf{v})=n(n-1) \bigl(1+\alpha ^{2}\bigr) \int _{M}f^{2}. $$

(28)

Also, on using

$$ \rho =-\alpha f, \qquad \lambda =(n-2) \bigl( 1+\alpha ^{2} \bigr),$$

we have

$$ n \int _{M} (f-\rho \alpha ) \bigl[(\lambda +1) f- \rho \alpha \bigr]=n(n-1) \bigl(1+\alpha ^{2}\bigr) \int _{M}f^{2} . $$

(29)

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

$$ \Delta f=-\frac{\tau }{n-1}f. $$

(30)

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

$$ \int _{M}S(\mathbf{v},\mathbf{v})\geq \frac{n-1}{n}\int _{M} ( \operatorname{div}\mathbf{v} ) ^{2} $$

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

$$ \Delta \rho =-\frac{\tau }{n-1}\rho . $$

(31)

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

$$ \operatorname{div}A\mathbf{v}=-nf\alpha +\rho \Vert A \Vert ^{2}+n \mathbf{v}(\alpha ), $$

and consequently, equation (19) implies

$$ \Delta \rho =nf\alpha -\rho \Vert A \Vert ^{2}-n \mathbf{v}(\alpha ). $$

(32)

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

$$ \rho ^{2} \bigl( \Vert A \Vert ^{2}-n\alpha ^{2} \bigr) =nf \rho \alpha +\frac{\tau }{n-1}\rho ^{2}-n\rho \mathbf{v}(\alpha )-n \rho ^{2}\alpha ^{2}. $$

Integrating the above equation and using Lemma 2.2, we get

$$ \int _{M}\rho ^{2} \bigl( \Vert A \Vert ^{2}-n \alpha ^{2} \bigr) = \int _{M} \biggl[ -n(n-1)f\rho \alpha +n(n-1) \rho ^{2}\alpha ^{2}+\frac{\tau }{n-1}\rho ^{2}-n\alpha g(A\mathbf{v}, \mathbf{v}) \biggr]. $$

(33)

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

$$ \int _{M} \Vert Av \Vert ^{2}= \int _{M} \Vert \operatorname{grad}\rho \Vert ^{2}=\frac{\tau }{n-1} \int _{M}\rho ^{2}. $$

(34)

Also, equation (13) gives

$$ \int _{M} \bigl[ \Vert Av \Vert ^{2}-n\alpha g(A \mathbf{v},\mathbf{v} \bigr] = \int _{M} \bigl[ (n-1) \Vert \mathbf{v} \Vert ^{2}-S(\mathbf{v},\mathbf{v}) \bigr] , $$

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

$$ \int _{M} \biggl[ \frac{\tau }{n-1}\rho ^{2}-n\alpha g(A \mathbf{v},\mathbf{v}) \biggr] = \int _{M} \bigl[ n(n-1) \bigl( f^{2}-f\rho \alpha \bigr) -S(\mathbf{v},\mathbf{v}) \bigr]. $$

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

$$ \int _{M}\rho ^{2} \bigl( \Vert A \Vert ^{2}-n \alpha ^{2} \bigr) = \int _{M} \bigl[ n(n-1) ( -f+\rho \alpha ) ^{2}-S( \mathbf{v},\mathbf{v}) \bigr]. $$

Now, using

$$ \operatorname{div}\mathbf{v}=n ( -f+\rho \alpha ) $$

in the above equation, we get

$$ \int _{M}\rho ^{2} \bigl( \Vert A \Vert ^{2}-n \alpha ^{2} \bigr) = \int _{M} \biggl[ \frac{(n-1)}{n} ( \operatorname{div} \mathbf{v} ) ^{2}-S(\mathbf{v},\mathbf{v}) \biggr]. $$

(35)

Using now the hypothesis

$$ \int _{M}S(\mathbf{v},\mathbf{v})\geq \frac{n-1}{n}\int _{M} ( \operatorname{div}\mathbf{v} ) ^{2} $$

in equation (35), we conclude

$$ \rho ^{2} \bigl( \Vert A \Vert ^{2}-n\alpha ^{2} \bigr) =0 . $$

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

$$ S(\mathbf{v},\mathbf{v})=(n-1) \bigl(1+\alpha ^{2}\bigr) \Vert \mathbf{v} \Vert ^{2} $$

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

$$ \int _{M}S(\mathbf{v},\mathbf{v})=n(n-1) \bigl(1+\alpha ^{2}\bigr) \int _{M}f^{2}. $$

(36)

Also, we have

$$\begin{aligned} \operatorname{div}\mathbf{v} =&n ( -f+\rho \alpha ) \\ =&n\bigl(1+\alpha ^{2}\bigr) (-f), \end{aligned}$$

and we derive

$$ \frac{n-1}{n} \int _{M} ( \operatorname{div}\mathbf{v} )^{2}=n(n-1) \bigl(1+ \alpha ^{2}\bigr) \int _{M}f^{2}. $$

(37)

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 $\mathbf{S}^{n}(1+\alpha ^{2})$. □