Geometric and topological rigidity for compact submanifolds of odd dimension (Q477125)

In 1986, Simon used rigidity theorem for closed minimal submanifolds in a sphere. Since then, several rigidity results for closed minimal submanifolds in a sphere were proved by Chern-do Carmo-Kobayashi, Lawson, Yau and others. In the present paper, the authors prove that if \(M^n\) (\(n\geq 5\)) is an odd-dimensional compact submanifold with parallel mean curvature in \(S^{n+p}\), with certain conditions on the Ricci curvature and mean curvature then \(M\) is a totally umbilical sphere. Moreover the authors prove that if \(M^n\) (\(n\geq 5\)) is an odd-dimensional compact submanifold in an (\(n+p\))-dimensional simply connected space form with constant curvature \(c\geq0\), under conditions on the Ricci curvature and mean curvature it is homeomorphic to a sphere. The results are interesting.