Focal Radius, Rigidity, and Lower Curvature Bounds

We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{\pi }{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective space with their standard metrics, provided $N$ is closed. Our results also hold for $k^{th}$--intermediate Ricci curvature, provided the submanifold has dimension $\geq k.$ Thus in a manifold with Ricci curvature $\geq n-1,$ all hypersurfaces have focal radius $\leq \frac{\pi }{2},$ and space forms are the only such manifolds where equality can occur, if the submanifold is closed. To prove these results, we develop a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation.Comment: The first part of the paper has been rewritten to simplify the proofs of the comparison theory for Wilking's transverse Jacobi equation. We have also corrected minor typos and reordered some of the material to simplify the readin

The Sub-Index of Critical Points of Distance Functions

We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.Comment: We corrected a mistake in the proof of Theorem 3.