Obstructions to nonpositive curvature for open manifolds

TL;DR

This paper investigates algebraic conditions on groups influencing nonpositive curvature in open manifolds, establishing structure theorems and demonstrating the existence of manifolds without nonpositive curvature metrics despite Euclidean universal covers.

Contribution

It introduces algebraic criteria for group actions on Hadamard manifolds and proves a structure theorem for certain nonpositively curved manifolds, also showing the existence of manifolds lacking such metrics.

Findings

Every open nonpositively curved K(G,1) manifold homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood.

Infinitely many open K(G,1) manifolds admit no complete nonpositively curved metric despite Euclidean universal cover.

An open contractible manifold W is Euclidean if and only if W×S¹ admits a nonpositive curvature metric.

Abstract

We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their universal cover is the Euclidean space. A sample application is that an open contractible manifold W is homeomorphic to a Euclidean space if and only if the product of W and a circle admits a complete Riemannian metric of nonpositive curvature.