  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
$k^{th}$--intermediate Ricci curvature that have a closed submanifold with
dimension $\geq k$ and infinite focal radius.
  To prove these results, we use the comparison lemma for Jacobi fields from
arXiv:1603.04050 that exploits Wilking's transverse Jacobi equation. The
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.
