Focal Radius, Rigidity, and Lower Curvature Bounds

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 $\ge k.$ Thus in a manifold with Ricci curvature $\ge n-1,$ all hypersurfaces have focal radius $\le \frac{\pi }{2},$ 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.

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 reading

References in corpus (4)

• Riemannian foliations of spheres
• How to lift positive Ricci curvature
• Low dimensional Singular Riemannian Foliations in spheres
• Clifford algebras and new singular Riemannian foliations in spheres

Cited by in corpus (8)

• Structure of Submetries
• Local symmetry rank bound for positive intermediate Ricci curvatures
• Torus actions on manifolds with positive intermediate Ricci curvature
• The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces
• Positive intermediate Ricci curvature with maximal symmetry rank
• Positive intermediate Ricci curvature on products of homogeneous spaces
• Infinite families of manifolds of positive $k^{\mathrm{th}}$-intermediate Ricci curvature with $k$ small
• Gehring Link Problem, Focal Radius and Over-torical width