Integral formulas and hyperspheres in a simply connected space form

Abstract

Abstract.Let M" denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic n + 1 space or in an open hemisphere of S"+ '. We show that if two consecutive mean curvatures of M are constant then M is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold M possesses a generalized position vector field; that is a vector field Y whose covariant derivative is at each point a multiple of the identity. In addition we prove that if M is complete and connected with the covariant derivative of Y exactly the identity at each point then M is isometric to Euclidean space.

Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications

“…On the other hand, in [9] Bivens proved that when the ambient space is the Euclidean space, the hyperbolic space, or the open hemisphere, then every immersed compact hypersurface with H r and H r+1 both constant must be a geodesic sphere. Related to this, Koh [22] has recently proved the same kind of result under the hypothesis that the ratio H s /H r is constant, 1 r < s n.…”

“…Higher-order Minkowski formulae for hypersurfaces were first obtained by Hsiung [20] in Euclidean space, and by Bivens [9] in the Euclidean sphere and hyperbolic space. These were generalized by Alencar and the third author [3] by using the (r + 1)-mean curvature linearized operator L r of the hypersurface.…”

A new characterization of geodesic spheres in the hyperbolic space

“…There are many works generalizing Theorem 1.1. For instance, In [20,21], Koh gave a new characterization of spheres in terms of the ratio of two mean curvature which generalized a previous result of Bivens [7]. Aledo-Alás-Romero [2] extended the result to compact space-like hypersurfaces with constant higher order mean curvature in de Sitter space.…”

A characterization of round spheres in space forms

“…Because of the difficulty of the above question, it is natural to attempt to obtain the rigidity of the sphere in R n+1 under geometric conditions stronger than H r be constant for some r. In this regard, Gardner [13] proved that if a compact oriented hypersurface M n in R n+1 has two consecutive mean curvatures H r and H r+1 constant, for some r = 1, ..., n−1, then it is a geodesic sphere. For generalizations of this result see [3,15,26].…”