# Geometry-Topology Seminar

From March 03, 2017 14:00 until March 03, 2017 15:00

Dr. Fred Wilhelm (UC Riverside) – *Jacobi Field Comparison*

Abstract:

Along a geodesic in a Riemannian manifold, the Jacobi fields encode how the pattern of nearby

geodesics diverge from the corresponding pattern in Euclidean space. In particular, complete

knowledge of a manifold's Jacobi fields completely determines its local geometry as well as

much of its algebraic topology.

As Jacobi fields are the solutions of a 2-nd order linear ODE that involves curvature, it is not

surprising that knowledge of curvature can be translated into knowledge of Jacobi fields. For

manifolds with curvature bounded from below there is a result along these lines that is known

as the Rauch Comparison Theorem. This classical theorem implies that in a manifold of

nonnegative curvature, the Jacobi fields (and their derivatives) are smaller than the

corresponding fields (and derivatives) in R^2. The comparison only works for a field whose

initial value or initial derivative is 0. This is a major drawback.

In most cases, the inequality is strict. When it is strict at some time, t1, examples suggest that it

typically is stricter at times t > t1. Unfortunately, other examples show that this is not always

the case.

Recently Guijarro and I remedied this situation by proving a comparison lemma. It shows that

when the Rauch inequality is strict for some field at some time, t1, the expected "even stricter"

estimate holds at future times for a field that is possibly different from the original one. The

proof exploits a powerful, but technical tool: Wilking's Transverse Jacobi equation.

Our comparison lemma has several applications including the following optimal finite-ness

theorem.

**Theorem:** (Guijarro-W.) Let M be a compact Riemannian manifold. Given D, r > 0 the class S

of closed Riemannian manifolds that can be isometrically embedded into M with focal radius r

and intrinsic diameter D is precompact in the C^1 topology.

In particular, S contains only finitely many diffeomorphism types.

I will give a gentle survey of the metric aspects of Riemannian geometry, as they relate to

Jacobi field comparison, with the goal of elucidating aspects of the statement and proof of this

theorem.

Friday, March 3rd

2:00 - 3:00 pm, MH 476

Please join us!