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^π ί (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*. The lift of a flat fc-torus to X is a /c-flat in X, i.

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... e. a complete totally geodesic and isometric embedding F: HS k -> X. We are interested in the opposite question: Does the existence of a k-flat in X imply the existence of a flat k-torus in M? The main result of this paper is an affirmative answer in the case that the flat has codimension ^ 2 and the metric is real analytic. Theorem 1. Let M be a complete real analytic n-dimensional Riemannian manifold with sectional curvature -b 2^K^O and offlnite volume. Let π: X -·» M be the universal covering space. If X contains a k-flat F, k^n -2 such that n(F) is contained in a compact subset of M, then M contains a totally geodesic flat immersed k-torus. The analyticity condition is crucial. If we assume M only to be smooth (i.e. C°°smooth) and complete of finite volume, then there are counterexamples: Theorem 2. For every integer n ^ 4, there exists a complete smooth n-dimensional manifold M with sectional curvature -b 2^K^O and finite volume, such that the universal covering X of M contains an (n -2)-flat F but there are no immersed flat tori in M. The flat F satisfies that n(F) is contained in a compact subset of M, where π: X -» M is the canonical projection.