Rigidity theorems of Clifford Torus

Luiz A M Sousa Jr

doi:10.1590/s0001-37652001000300003

Abstract

Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus <img src="http:/img/fbpe/aabc/v73n3/03ab.gif" alt="03ab.gif (725 bytes)" align="middle">.

Highlights

Let M be an n-dimensional hypersurface in a unit sphere Sn+1
In this note we give a sketch of the proof of the following results
If M has (n − 1) principal curvatures with the same signal everywhere, M is isometric to a Clifford Torus S1

Summary

Introduction

Let M be an n-dimensional hypersurface in a unit sphere Sn+1. We choose a local ortonormal frame field {e1, ..., en+1} in Sn+1, so that, restricted to M, e1, ..., en are tangent to M. Let w1, ..., wn+1 denote the dual co-frame field in Sn+1. It follows from Cartan’s Lemma that wn+1,i = hij wj , hij = hji . The second fundamental form h and the mean curvature H of M are defined by

Results

Conclusion