Rigidity of compact submanifolds in a unit sphere
Let \(M\) be a compact Riemannian manifold of dimension \(n\) immersed in a unit sphere \(S^{n + p}\) of dimension \(n + p\). Let \(S\) be the square of the length of the second fundamental form. The study of upper bounds for \(S\) in connection with some properties of \(M\) is an interesting problem. The aim of the present paper is to give some results in this direction, which can be stated as follows. (I) Let \(M^n\) be an \(n\)-dimensional compact submanifold in \(S^{n + p}\) with non-zero parallel mean curvature. If either of the following conditions is satisfied, then \(M^n\) lies in a totally geodesic \(S^{n+1}\): (i) \(S \leq \min \left \{ {2 \over 3} n,\;{2n \over 1 + \sqrt{n \over 2}}\right\}\), \(p > 2\) and \(n \neq 8\), (ii) \(S \leq \min \left \{ {n \over 2 - {1\over p - 1}} , {2n \over 1 + \sqrt {n\over 2}} \right \}\), \(p > 1\) and \((n,p) \neq (8, 3)\). (II) Let \(M^n\) be an \(n\)-dimensional \((2 \leq n \leq 7)\) compact submanifold in \(S^{n + p}\) with \(p > 2\) which has non-zero parallel mean curvature. If \(S \leq (2/3)n\), then \(M^n\) is totally umbilical. (III) Let \(M^n\) be a compact minimal submanifold in the sphere \(S^{n + p}\) with \(p > 1\). Suppose the sectional curvatures of \(M^n\) are everywhere not less than \((1/2 - 1/3p)\). Then either \(M^n\) is a totally geodesic sphere or the Veronese surface in \(S^4\).
scientific article; zbMATH DE number 815800