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 \(\mathbf{S}^{n}(1+\alpha ^{2})\). □