Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature
The author determines the configuration of flat, totally geodesic submanifolds in a compact, real analytic manifold M of sectional curvature \(K\leq 0\) and dimension 3 or 4. This configuration depends only on the fundamental group of M (Corollary 2 below). Let \(\tilde M\) denote a complete, simply connected, real analytic manifold with sectional curvature \(K\leq 0\), and let \(M=\tilde M/\Gamma\) denote a compact quotient manifold of M, where \(\Gamma\) \(\subseteq I(\tilde M)\) is a discrete group isomorphic to the fundamental group of M. A k-flat \(F\subseteq \tilde M\) is a complete, flat, imbedded, totally geodesic submanifold of \(\tilde M.\) A complete, totally geodesic submanifold X of \(\tilde M\) is said to be closed (relative to M) if the image of X under the projection \(\pi: \tilde M\to M\) is compact. The following two questions are basic: 1) If \(\tilde M\) admits a k-flat F does it also admit a (possibly different) closed k-flat \(F'?\) 2) Are the closed k-flats of \(\tilde M\) dense in the space of all k-flats of \(\tilde M\)? These questions are unresolved for \(C^{\infty}\) manifolds of arbitrary dimension. The author gives an affirmative answer to both questions for real analytic manifolds of dimension 3 or 4. Theorem 1. Let \(M=\tilde M/\Gamma\) be a compact 3-dimensional real analytic manifold with sectional curvature \(K\leq 0\). If \(\tilde M\) is not a product \(\tilde M'\times R\), then there are (modulo \(\Gamma)\) only finitely many 2-flats in \(\tilde M.\) Each 2-flat is closed and any two different 2-flats \(F,F'\) in \(\tilde M\) are disjoint. A nonpositively curved manifold M is said to be of higher rank if every geodesic of the universal cover \(\tilde M\) is contained in a k-flat for some integer \(k\geq 2\). Theorem 2. Let \(M=\tilde M/\Gamma\) be a compact 4- dimensional real analytic manifold with sectional curvature \(K\leq 0\). Then there are (modulo \(\Gamma)\) only finitely many totally geodesic submanifolds of higher rank in \(\tilde M,\) and each such submanifold is closed. Corollary 1. Let \(M=\tilde M/\Gamma\) be as in Theorem 2. Then the closed 2-flats are dense in the set of all 2-flats of \(\tilde M.\) Corollary 2. Let \(M_ 1=\tilde M_ 1/\Gamma_ 1\) and \(M_ 2=\tilde M_ 2/\Gamma_ 2\) be as in Theorem 1 or 2 and let \(\theta: \Gamma_ 1\to \Gamma_ 2\) be an isomorphism. Then there is a bijection \({\bar \theta}\) between the totally geodesic higher rank submanifolds of \(\tilde M_ 1\) and \(\tilde M_ 2\) such that \({\bar \theta}(\gamma W)=\theta (\gamma){\bar \theta}(W)\) for all \(\gamma \in \Gamma_ 1\). Moreover if higher rank submanifolds \(W_ 1\) and \(W_ 2\) intersect, then so do their images under \({\bar \theta}\).
scientific article; zbMATH DE number 4111494