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 2π. 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 kth--intermediate Ricci curvature, provided
the submanifold has dimension ≥k. Thus in a manifold with Ricci curvature
≥n−1, all hypersurfaces have focal radius ≤2π, 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
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