Extrinsic spheres in a real space form

Abstract

Let M be an n-dimensional orientable compact hypersurface in an (n+1)-dimensional real space form M(c), n ≥2. If the lengths ∥ℝ∥, ∥double-struck A∥ and ∥∇α∥ of the curvature tensor field R, the shape operator A, the gradient ∇α of the mean curvature α and the scalar curvature S of the hypersurface M satisfy the inequality (Equation Presented) where δ = min Ric = min (Equation Presented) Ricp(v), Ric is Ricci curvature of the hypersurface, then it is shown that M is an extrinsic sphere in M(c). In particular we deduce that the condition 1/2 ∥R∥2 ≤ δ ∥A∥2 - n(n - 1) ∥∇α∥2 characterizes spheres in the Euclidean space Rn+l among the compact orientable hypersurfaces whose Ricci curvatures are bounded below by a constant δ > 0.