Flats in spaces with convex geodesic bicombings

TL;DR

This paper extends classical results about flat embeddings and hyperbolicity from nonpositive curvature spaces to more general metric spaces with convex geodesic bicombings, broadening the understanding of geometric structures in these spaces.

Contribution

It generalizes key theorems like the Flat Torus Theorem and Gromov's hyperbolicity criterion to spaces with convex geodesic bicombings, even with non-unique geodesics.

Findings

Validates flat embedding results in broader metric spaces

Extends Gromov's hyperbolicity criterion beyond classical settings

Generalizes Bowditch's results for Busemann spaces

Abstract

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov's hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.