Restrictions on submanifolds via focal radius bounds

We give an optimal estimate for the norm of any submanifold’s second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold.

This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,\alpha}$ compactness result for submanifolds, as well as a “soul-type” structure theorem for manifolds with nonnegative $k^{\textrm{th}}$-intermediate Ricci curvature that have a closed submanifold with dimension $\geq k$ and infinite focal radius.

To prove these results, we use a new comparison lemma for Jacobi fields from [18] that exploits Wilking’s transverse Jacobi equation. The new comparison lemma also yields new information about group actions, Riemannian submersions, and submetries, including generalizations to intermediate Ricci curvature of results of Chen and Grove. None of these results can be obtained with just classical Riccati comparison (see Subsection 3.1 for details.)

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The first author’s work was supported by research grants MTM2011-22612, MTM2014-57769-3-P, and MTM2017-85934-C3-2-P from the MINECO, and by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).

The second author’s work was supported by a grant from the Simons Foundation (#358068, Frederick Wilhelm).

Received 30 April 2018

Accepted 20 February 2019

Published 8 April 2020