Fat flats in rank one manifolds

TL;DR

This paper explores the geometric and dynamical implications of fat flats in rank one non-positively curved manifolds, revealing new constructions and a closing theorem that influence the structure and abundance of closed geodesics.

Contribution

It introduces the concept of fat flats in rank one manifolds, constructs examples with specific properties, and proves a closing theorem linking fat flats to closed geodesics.

Findings

Existence of rank 1 manifolds with fat 1-flats and countably many closed geodesics.

Construction of metrics on covers of hyperbolic manifolds with flat neighborhoods.

A closing theorem showing fat flats imply totally geodesic flat submanifolds and uncountably many geodesics.

Abstract

We study closed non-positively curved Riemannian manifolds $M$ which admit `fat $k$-flats': that is, the universal cover $\tilde M$ contains a positive radius neighborhood of a $k$-flat on which the sectional curvatures are identically zero. We investigate how the fat $k$-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank $1$ non-positively curved manifolds with a fat $1$-flat which corresponds to a twisted cylindrical neighborhood of a geodesic on $M$. As a result, $M$ contains an embedded closed geodesic with a flat neighborhood, but $M$ nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is to prove a closing theorem for fat flats, which implies that a manifold $M$ with a fat $k$-flat contains an immersed, totally geodesic $k$-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when $k \geq 2$. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.