Codimension one tori in manifolds of nonpositive curvature
The main result of this article is the following Theorem: Let M be a complete, \(C^{\infty}\), Riemannian manifold with finite volume and sectional curvature satisfying \(-b^ 2\leq K\leq 0\). If the universal Riemannian cover \(\tilde M\) contains a complete, flat, totally geodesic submanifold of codimension 1 (=codimension 1 flat), then M contains a complete, flat, totally geodesic immersed torus of codimension 1. The author has proved an even stronger result if the manifold M is real analytic [J. Reine Angew. Math. 390, 32-46 (1988; Zbl 0641.53042)]: either \(\tilde M\) is isometric to \(M^*\times {\mathbb{R}}^{n-2}\), where dim \(M^*=2\), or every codimension 1 flat in \(\tilde M\) projects to an immersed, flat, \((n-1)\)-torus in M. In that article the author also shows that the attempt to extend the theorem of this article to the case of codimension 2 flats fails for noncompact manifolds M of finite volume and bounded nonpositive curvature. Recently the author in joint work with V. Bangert has proved the following remarkable result: Theorem. Let M be a compact real analytic manifold with sectional curvature \(K\leq 0\). If the universal Riemannian cover \(\tilde M\) admits a k-flat of arbitrary codimension, where \(k\geq 2\), then M admits an isometrically immersed, flat, k-torus. It is still an open question whether this result remains true in the \(C^{\infty}\) case. As a consequence of the result just stated one has the following Corollary. Let M be a compact, real analytic manifold with sectional curvature \(K\leq 0\). Then the universal cover \(\tilde M\) satisfies the visibility axiom if and only if every Abelian subgroup of the fundamental group of M is infinite cyclic.
scientific article; zbMATH DE number 4144873