Geometric and topological rigidity for compact submanifolds of odd dimension

We investigate rigidity problems for odd-dimensional compact submanifolds. We show that if M ⁿ (n ? 5) is an odd-dimensional compact submanifold with parallel mean curvature in S ^n+p , and if Ric _M > (n?2? \(\tfrac{1} {n}\) )(1+H ²) and H < δ _n , where δ _n is an explicit positive constant depending only on n, then M is a totally umbilical sphere. Here H is the mean curvature of M. Moreover, we prove that if M ⁿ (n ? 5) is an odd-dimensional compact submanifold in the space form F ^n+p (c) with c ? 0, and if Ric _M > (n?2?? _n )(c+H ²), where ? _n is an explicit positive constant depending only on n, then M is homeomorphic to a sphere.