Geometry of the spectrum (Seattle, WA, 1993) (19940101), 225-229.
Subject
58 Global analysis, analysis on manifolds -- 58F Ordinary differential equations on manifolds; dynamical systems 58F19 Eigenvalue and spectral problems
Let $M$ and $N$ be two closed Riemannian manifolds of negative curvature and let $SM$ and $SN$ denote the unit sphere bundles of $M$ and $N$, respectively. Let $G_{t}^{M}$ denote the geodesic flow of $M$. Suppose $F\colon SM\rightarrow SN$ is a homeomorphism that commutes with the geodesic flows of $M$ and $N$. Then the author shows that there exists $t\in {\bf R}$ such that the map given by $F_{1}=F\circ G_{t}^{M}$ satisfies $F_{1}(v)=-F_{1}(-v)$ for all $v\in SM$. As a corollary it follows that if $M$ and $N$ are in addition locally symmetric, then $F$ must have the form $dI\circ G_{t}^{M}$, where $I\colon M\rightarrow N$ is an isometry and $t$ is some real number.