Noncritical maps on geodesically complete spaces with curvature bounded above

TL;DR

This paper investigates the regularity of distance maps on geodesically complete spaces with curvature bounds, establishing their local fibration property and deriving a sphere theorem for CAT(1) spaces.

Contribution

It introduces a new regularity concept for distance maps in spaces with curvature bounds and proves their local fibration property, extending geometric regularity theory.

Findings

Distance maps are locally Hurewicz fibrations.

Regularity dual to Perelman's concept in Alexandrov spaces.

Sphere theorem for geodesically complete CAT(1) spaces.

Abstract

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of Perelman's regularity in the geometry of Alexandrov spaces with curvature bounded below. As a corollary we obtain a sphere theorem for geodesically complete CAT(1) spaces.