Symmetry diffeomorphism group of a manifold of nonpositive curvature

Abstract:

Let $\tilde M$ denote a complete simply connected manifold of nonpositive sectional curvature. For each point $p \in \tilde M$ let ${s_p}$ denote the diffeomorphism of $\tilde M$ that fixes $p$ and reverses all geodesics through $p$. The symmetry diffeomorphism group ${G^{\ast }}$ generated by all diffeomorphisms $\{ {s_p}: p \in \tilde M\}$ extends naturally to group of homeomorphisms of the boundary sphere $\tilde M(\infty )$. A subset $X$ of $\tilde M(\infty )$ is called involutive if it is invariant under ${G^{\ast }}$. Theorem. Let $X \subseteq \tilde M(\infty )$ be a proper, closed involutive subset. For each point $p \in \tilde M$ let $N(p)$ denote the linear span in ${T_p}\tilde M$ of those vectors at $p$ that are tangent to a geodesic $\gamma$ whose asymptotic equivalence class $\gamma (\infty )$ belongs to $X$. If $N(p)$ is a proper subspace of ${T_p}\tilde M$ for some point $p \in \tilde M$, then $\tilde M$ splits as a Riemannian product ${\tilde M_1} \times {\tilde M_2}$ such that $N$ is the distribution of $\tilde M$ induced by ${\tilde M_1}$. 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 $\tilde M$ is irreducible and $\tilde M(\infty )$ admits a proper, closed involutive subset $X$, then $\tilde M$ is isometric to a symmetric space of noncompact type and rank $k \geqslant 2$.