Rigidity of compact minimal submanifolds in a sphere
In this paper we study n-dimensional compact minimal submanifolds in S 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 with flat normal connection satisfies n(n − p − 1) ≤ S ≤ n(n − 2) (cf. Theorem 4.1). Since for a compact hypersurface M of S the normal connection is flat, we use the above estimate for a scalar curvature S of a non-negatively curved minimal hypersurface M in S to infer that either M is totally geodesic or else it is isometric
Curvature pinching for three-dimensional minimal submanifolds in a sphere
In this paper, some pinching theorems for the Ricci curvature and the scalar curvature of three-dimensional compact minimal submanifolds in a sphere are given.
Totally real submanifolds in a 6-sphere
A 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. We investigate 3-dimensional totally real submanifolds in 56 and classify 3-dimensional totally real submanifolds of constant sectional curvature.