Differentiable rigidity of hypersurface in space forms

Abstract

Let M be a compact n-dimensional hypersurface in the space form \({F^{n+1}(c)}\) where n = 3 or 4. We prove that if the scalar curvature of M is positive and the integral of the length of the second fundamental form of M satisfies certain inequality, then M must be diffeomorphic to the spherical space form. On the other hand, we prove that if M is homeomorphic to 2-dimensional sphere \({\mathbb{S}^2}\) and f1, f2, f3 are first eigenfunctions of M such that \({\sum\nolimits_{i=1}^3|\nabla f_{i}|^2}\) is a constant, then M is isometric to a sphere with constant curvature.