Remarks on sphere-type theorems

We prove if $M$ is a complete Riemannian manifold with an embedded totally geodesic compact hypersurface $N$ such that $M$ has nonnegative sectional curvature, and the sectional curvature of $M$ is strictly positive in a neighborhood of $N$, then the pair $(M,N)$ is diffeomorphic to the pair $(S^n,S^{n-1})/\pi _1(M)$. This result gives an affirmative answer to a question of H. Wu in the case when $M$ is compact and simply connected.