Focal Radius, Rigidity, and Lower Curvature Bounds

TL;DR

This paper establishes upper bounds on the focal radius of submanifolds in positively curved manifolds, characterizes the cases of equality, and introduces a new Jacobi field comparison lemma to prove these results.

Contribution

It provides new bounds on focal radii in manifolds with curvature constraints and characterizes the geometric structure when bounds are achieved, using a novel Jacobi field comparison technique.

Findings

Focal radius of submanifolds in manifolds with curvature ≥ 1 does not exceed π/2.

Equality cases imply the submanifold is totally geodesic and the ambient space is a sphere or projective space.

Results extend to intermediate Ricci curvature and characterize space forms as unique cases for equality.

Abstract

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.