Obstructions to nonpositive curvature for open manifolds (Q2922882)

A connected smooth \(n\)-manifold is said to be covered by \({\mathbb R}^n\) if its universal cover is diffeomorphic to \({\mathbb R}^n\), and a manifold is covered by \({\mathbb R}\times {\mathbb R}^{n-1}\) if it is diffeomorphic to the product of \({\mathbb R}\) and a manifold covered by \({\mathbb R}^{n-1}\). Let \(G\) be the deck transformation group acting isometrically on the universal cover \(X\). It is well known that if \(G\) fixes a Busemann function, i.e., \(G\) fixes a point at infinity and stabilizes each horosphere centered at the point, then \(X/G\) is covered by \({\mathbb R}\times {\mathbb R}^{n-1}\).NEWLINENEWLINEIn this paper, the author justifies that the property `\(G\) fixes a Busemann function' can be forced by purely algebraic assumptions on \(G\), and studies algebraic conditions on a group \(G\) under which every properly discontinuous, isometric \(G\)-action on a Hadamard manifold has a \(G\)-invariant Busemann function. Also the author proves that, for such \(G\), every open complete nonpositively curved Riemannian \(K(G, 1)\)-manifold that is homotopy equivalent to a finite complex of codimension at least \(3\) is an open regular neighborhood of a subcomplex of the same codimension, and 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. As an application, the author shows that an open contractible manifold \(W\) is homeomorphic to a Euclidean space if and only if \(W\times S^1\) admits a complete Riemannian metric of nonpositive curvature.