Manifolds all of whose flats are closed (Q1383657)

A compact Riemannian manifold is of higher rank if any geodesic lies in a flat totally geodesic submanifold (``flat'') of dimension \(\geq 2\). If such a manifold \(M\) has sectional curvature \(K \leq 0\), then the well-known rank rigidity theorem mainly due to W. Ballmann says that \(M\) is locally symmetric. Without the curvature assumption, this is wrong; in particular, it fails for \(K \geq 0\): the easiest counterexample is a normally homogeneous space \(G/T'\) where \(G\) is a compact Lie group and \(T'\) a subtorus of a maximal torus with codimension \(\geq 2\). However, in the counterexamples almost all flats are noncompact. In the present paper, the following main theorems are shown (in fact, the assumptions made are slightly more general). Theorem If \(M\) is compact and locally irreducible such that any geodesic lies in a compact flat of dimension \(\geq 2\), then \(M\) is locally symmetric. Corollary. A compact simply connected irreducible analytic manifold all of whose geodesics lie in a closed flat is either symmetric or a manifold all of whose geodesics are closed. The central idea of the proof is as follows. By the holonomy theorem of Berger and Simons it suffices to show that the local holonomy group at any point \(p\) in some open dense subset does not act transitively on the unit sphere of \(T_pM\). Assume the contrary and consider a parallel unit vector field \(v(t)\) along a short piecewise smooth closed path \(c(t)\) starting and ending at \(p\). One finds a flat \(k\)-torus \(T\) (with \(k \geq 2\)) and a piecewise smooth family of totally geodesic affine immersions \(f_t : T \to M\) passing through \(c(t)\) and being tangential to \(v(t)\). Thus \({\partial \over \partial t}f_t\) is a Jacobi field along \(f_t\). As in the case where \(f_t\) is a geodesic, this splits into a tangent and a normal component with respect to \(f_t\), where the tangent component is affine and hence parallel by periodicity. Let \(v_0\) be the tangent vector of \(T\) with \(df_0.v_0 = v(0)\). Then \({\partial \over \partial t}\langle df_t.v_0,v(t)\rangle = \langle{\partial\over\partial t} df_t.v_0,v(t)\rangle = 0\) since \({\partial\over\partial t}df_t.v_0 = \nabla_{v_0}({\partial\over\partial t}f_t)\). Hence \(v(1)\) satisfies the equality \[ \langle df_1.v_0,v(1)\rangle = \langle df_0.v_0,v(0)\rangle=1.\tag{\(*\)} \] This shows that \(v(1)\) cannot be an arbitrary unit vector near \(v(0)\). In fact, we may assume that \(v(1)\) spans the immersed torus \(F_1 = f_1(T)\) in the sense that \(F_1\) is the closure of the geodesic \(\gamma_{v(1)}\) (the set of vectors failing to satisfy this property is ``small''). Hence \(F_1\) is the only \(k\)-flat being tangential to \(v(1)\). But up to translations, there are only countably many affine parametrizations \(f : T \to F_1\) and thus only countably many vectors of the form \(df.v_0\) in \(T_pF_1\). Thus by \((*)\), the vector \(v(1)\) lies in a countable union of hyperplanes in \(T_pF_1\) which contradicts our assumption. The compactness assumption is heavily used in the proof. As was recently shown by E. Samiou, it can be replaced by the assumption that the flats are extrinsically homogeneous and isometric.