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Existence of immersed tori in manifolds of nonpositive curvature.

In this paper we study complete Riemannian manifolds of nonpositive sectional curvature K^O. The universal covering X of such a manifold M is diffeomorphic to euclidean space and M can be identified with Χ/Γ, where Γ^πί(Μ) is the group of decktransformations. The flat torus theorem [GW], [LY] states: If M is compact, then M contains a totally geodesic immersed k-dimensional flat torus if and only if Γ contains a subgroup isomorphic to Z*.