Symmetry diffeomorphism group of a manifold of nonpositive curvature

Abstract

Let M ~ \tilde M denote a complete simply connected manifold of nonpositive sectional curvature. For each point p ∈ M ~ p \in \tilde M let s p {s_p} denote the diffeomorphism of M ~ \tilde M that fixes p p and reverses all geodesics through p p . The symmetry diffeomorphism group G ∗ {G^{\ast }} generated by all diffeomorphisms { s p : p ∈ M ~ } \{ {s_p}:\,p \in \tilde M\} extends naturally to group of homeomorphisms of the boundary sphere M ~ ( ∞ ) \tilde M(\infty ) . A subset X X of M ~ ( ∞ ) \tilde M(\infty ) is called involutive if it is invariant under G ∗ {G^{\ast }} . Theorem. Let X ⊆ M ~ ( ∞ ) X \subseteq \tilde M(\infty ) be a proper, closed involutive subset. For each point p ∈ M ~ p \in \tilde M let N ( p ) N(p) denote the linear span in T p M ~ {T_p}\tilde M of those vectors at p p that are tangent to a geodesic γ \gamma whose asymptotic equivalence class γ ( ∞ ) \gamma (\infty ) belongs to X X . If N ( p ) N(p) is a proper subspace of T p M ~ {T_p}\tilde M for some point p ∈ M ~ p \in \tilde M , then M ~ \tilde M splits as a Riemannian product M ~ 1 × M ~ 2 {\tilde M_1} \times {\tilde M_2} such that N N is the distribution of M ~ \tilde M induced by M ~ 1 {\tilde M_1} . This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if M ~ \tilde M is irreducible and M ~ ( ∞ ) \tilde M(\infty ) admits a proper, closed involutive subset X X , then M ~ \tilde M is isometric to a symmetric space of noncompact type and rank k ⩾ 2 k \geqslant 2 .