Focal radius, rigidity, and lower curvature bounds

Abstract

We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point.

Using it, 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 π 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 that N is closed.

Our results also hold for k th intermediate Ricci curvature, provided that the submanifold has dimension ⩾ k . Thus, in a manifold with Ricci curvature ⩾ n − 1 , all hypersurfaces have focal radius ⩽ π 2 , and space forms are the only such manifolds where equality can occur, if the submanifold is closed.

Example 4.38 and Remark 5.4 show that our results cannot be proven using standard Riccati or Jacobi comparison techniques