  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.
