Geodesically complete spaces with an upper curvature bound

Abstract

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.2010 Mathematics Subject Classification. 53C20, 53C21, 53C23.

Improvements of upper curvature bounds

“…The main result of this note confirms the expectation that in local considerations the value of the upper curvature bound does not matter: In particular, in many questions concerning only local topological properties of CAT(κ) spaces, like most of [Kle99], [LN19], [LN18], one may always assume κ to be −1.…”

“…Moreover, the tangent spaces at any point y ∈ O with respect to both metrics d and d ′ are isometric. This condition implies that (O, d ′ ) is geodesically complete if (O, d) is locally geodesically complete, see [LN19].…”

Open and discrete maps with piecewise linear branch set images are piecewise linear maps

“…67] for a related claim in the smooth setting. Compare also to the Perelman stability theorem in [22] (see also [15]) from which a similar result could be deduced in the smooth setting.…”

“…Rather, we rely here on the fact that in manifolds with positive injectivity radius, homotopy arguments can be essentially reduced to discrete homotopy. Note that the above-mentioned Perelman stability theorem [22] requires the assumption of a lower bound to the Ricci curvature, and such a lower bound also gives rise to a lower bound for the injectivity radius of a closed Riemannian manifold. Definition 5.1.…”

Quantitative Estimates on the Singular Sets of Alexandrov Spaces

“…In [9], similar results as Corollary 1.4 and Theorem 1.6 are also proved for geodesically complete spaces with upper curvature bounds.…”