Rigidity of compact minimal submanifolds in a sphere

Sharief Deshmukh

Vol. 193 (2000), No. 1, 31–44

DOI: 10.2140/pjm.2000.193.31

Abstract

In this paper we study n-dimensional compact minimal submanifolds in S^n+p with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. Theorem 3.1 and Corollary 4.1). We also show that the scalar curvature S of a non-totally geodesic n-dimensional non-negatively curved minimal submanifold in S^n+p with flat normal connection satisfies n(n − p − 1) ≤ S ≤ n(n − 2) (cf. Theorem 4.1). Since for a compact hypersurface M of Sⁿ⁺¹ the normal connection is flat, we use the above estimate for a scalar curvature S of a non-negatively curved minimal hypersurface M in Sⁿ⁺¹ to infer that either M is totally geodesic or else it is isometric to the hypersurface S^m× S^n−m. As a consequence this result, we conclude that the only non-negatively curved compact minimal hypersurfaces in Sⁿ⁺¹ which are diffeomorphic to Sⁿ is totally geodesic sphere.

Milestones

Received: 15 July 1998

Published: 1 March 2000