Rigidity of compact minimal submanifolds in a sphere
Let \(M^n\) be a compact minimal \(n\)-dimensional submanifold of the unit sphere \(S^{n+p}\). The author focusses on pinching conditions for the scalar curvature \(\tau\) of \(M\) which imly that \(M\) is a totally geodesic submanifold. First he proves that this is always the case when \(\tau> n(n-2)\) and \(p\leq 2\). To obtain a similar result for \(p>2\) he needs an additional condition on the curvature tensor of the normal connection (for example, he considers the case of a flat normal condition). Furthermore, he proves that when \(M\) is nonnegatively curved with flat normal connection, then \(M\) is totally geodesic or else \(\tau\) satisfies \(n(n-p-1)\leq\tau\leq n(n-2)\). From these results he also derives several properties for hypersurfaces \(M\), in which case the normal connection is automatically flat. For example, when \(M\) is a nonnegatively curved compact minimal hypersurface \(M\) in \(S^{n+1}\), then it is totally geodesic or it is isometric to the hypersurface \(S^m(\sqrt{{m\over n}})\times S^{n-m}(\sqrt{{n-m\over n}})\) (for which the square of the length of the second fundamental form equals \(n\)) which implies that if \(M\) is a nonnegatively curved compact minimal hypersurface in \(S^{n+1}\) which is diffeomorphic to \(S^n\), then it is a totally geodesic hypersurface.
scientific article; zbMATH DE number 1463777