Rank and symmetry of Riemannian manifolds
Let \(M\) be a complete irreducible Riemannian manifold. A \(k\)-flat in \(M\) is a complete connected flat totally geodesic immersed submanifold in \(M\) of dimension \(k\). The rank of \(M\) is the maximal dimension \(k\) such that every geodesic in \(M\) lies in a \(k\)-flat. Examples of manifolds of rank \(k\) are locally symmetric spaces of rank \(k\). There is a conjecture that these are the only examples for \(k \geq 2\), which is open for the case that \(M\) has sectional curvature \(K \leq 0\). Concerning this conjecture, the authors give a partial positive answer: If \(M\) is a complete irreducible Riemannian manifold of rank \(k\) satisfying the condition that there is a \(k\)-flat \(F\) in \(M\) and a Lie subgroup \(G\) of the full isometry group \(I(M)\) of \(M\) with bounded Killing fields such that \(G(TF) = TM\), then \(M\) is locally symmetric of rank \(k\). Moreover, if any \(k\)-flat is one-to-one immersed and intrinsically symmetric, then \(M\) is globally symmetric. This result is an improvement of the recent result by \textit{E. Heintze, R. Palais, C.-L. Terng} and \textit{G. Thorbergsson} [J. Reine Angew. Math. 454, 163-179 (1994; Zbl 0804.53074)]. Further, for the case \(k = 1\), the authors prove that a complete Riemannian manifold \(M\), such that \(I(M)\) acts transitively on the set of geodesics, is globally symmetric.
scientific article; zbMATH DE number 718593