Intrinsic Rank in CAT(0) Spaces

2022/05/09 by Ontaneda, Pedro, Ricks, Russell
#Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2205.04377

Abstract

Let X be a proper, geodesically complete CAT(0) space which satisfies Chen and Eberlein's duality condition. We show the existence of a strong notion of rank for X by proving that the parallel sets P_v of geodesics v in X are generically flat. More precisely, let GX be the space of parametrized unit-speed geodesics in X. There is a unique k and a dense G_δ set A in GX such that P_v is isometric to flat Euclidean space ℝ^k, for all v ∈ A. It follows that ℝ^k isometrically embeds in P_v for every v ∈ GX.

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