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

We use a nontrivial concircular vector field u on the unit sphere Sn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{S}^{n+1}$\end{document} in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere Sn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{S}^{n+1}$\end{document} 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 Sn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{S}^{n+1}$\end{document}. 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 Sn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{S}^{n+1}$\end{document}.


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][37][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 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 ∈ X(M), where X(M) is the Lie algebra of smooth vector fields on M. Similarly, the rough Laplace operator on the Riemannian manifold (M, g), : where ∇ is the Riemannian connection and {e 1 , . . . , 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 : X(M) → 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, ξ , λ), then ξ = 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 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 S n+1 as hypersurface of the Euclidean space R n+2 with unit normal ξ and shape operator B = -√ cI, where I denotes the identity operator. For the constant vector field − → a = ∂ ∂x 1 on the Euclidean space R n+2 , where x 1 , . . . , x n+2 are Euclidean coordinates on R n+2 , we denote by u the tangential projection of − → a on the unit sphere S n+1 . Then we have , is the Euclidean metric on R n+2 . Taking covariant derivative in the above equation with respect to a vector field X on the unit sphere S n+1 and using Gauss-Weingarten formulae for hypersurface, we conclude where ∇ is the Riemannian connection on the unit sphere S n+1 with respect to the canonical metric g and grad f is the gradient of the smooth function f on S n+1 . Thus, u is a concircular vector field on the unit sphere S n+1 . Now consider the small sphere S n ( 1 c 2 ) defined by Then it follows that S n ( 1 c 2 ) is a hypersurface of the unit sphere 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 S n ( 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 S n ( 1 c 2 ) is given by where α is the mean curvature of the hypersurface S n ( 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 S n ( 1 c 2 ) and define ρ = g(u, N). Then we have However, we can easily see using the definitions of u and N that where f is the restriction of f to S n ( 1 c 2 ). Thus, ρ = -αf . 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 ∈ X(S n ( 1 c 2 )). Also, we have grad f = v. Thus, the rough Laplace operator 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 S n ( 1 c 2 ) is given by Thus, we observe that the vector field v on the small sphere S n ( 1 c 2 ) satisfies v = (n -2) 1 + α 2 v.
Also, using equation (8), we see that the Hessian of ρ is given by for X, Y ∈ X(S n ( 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 S n ( 1 c 2 ) satisfies the Fischer-Marsden equation Thus, in view of equations (9) and (10), the small sphere S n ( 1 c 2 ) admits a vector field v that is an eigenvector of the de-Rham Laplace operator with eigenvalue (n -2)(1 + α 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 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 grad ρ = -Av 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).

Preliminaries
Let M be an orientable hypersurface of the unit sphere S n+1 with unit normal vector field N and shape operator A. We denote the canonical metric on 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 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 τ = n(n -1) + n 2 α 2 -A 2 , where X, Y , Z ∈ X(M) and α = 1 n Tr A is the mean curvature of the hypersurface M and A 2 = Tr A 2 . The Codazzi equation of hypersurface gives where Taking a local orthonormal frame {e 1 , . . . , e n } on the hypersurface M, equation (15) yields Let u be the concircular vector field on the unit sphere S n+1 considered in the previous section, which satisfies equation (3), where f is the function defined on S n+1 by f = − → a , ξ . We denote the restriction of f 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 f , i.e., 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 Proof Using equation (19), we have div v = n(-f + ρα), and using equation (18), we get div(f v) = v 2 + nf (-f + ρα).
Integrating the above equation, we get the result. Proof Note that we have div α(ρv) = ρv(α) + α div(ρv) Integrating this equation and using the second equation in (19), we get the result.

Characterizations of small spheres
Let u be the concircular vector field on the unit sphere S n+1 and M be its orientable nontotally 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 S n+1 . Proof 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 Therefore, we derive M -n(n + λ)f 2 + n(2n + λ)f ραn 2 ρ 2 α 2 + 2S(v, v) = 0.
Note that equation (18) implies and Bochner's formula gives Using equation (18), we have and Hence we derive Thus, from equation (24) Combining equations (23) and (25) (retaining out of 2S(v, v) one term in (24)), we get However, as the support function ρ = 0, we get A 2 = nα 2 , and this equality in view of Schwartz's inequality holds if and only if Using a local orthonormal frame {e 1 , . . . , e n } in the above equation, we get n i=1 (∇A)(e i , e i ) = grad α, and combining the above equation with equation (16), we get (n -1) grad α = 0.
As n ≥ 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 S n (1 + α 2 ).