Gromov’s Amenable Localization and Geodesic Flows

Abstract

Abstract Let M be a compact smooth Riemannian n-manifold with boundary. We combine Gromov’s amenable localization technique with the Poincaré duality to study the traversally generic geodesic flows on SM, the space of the spherical tangent bundle. Such flows generate stratifications of SM, governed by rich universal combinatorics. The stratification reflects the ways in which the flow trajectories are tangent to the boundary

$$\partial (SM)$$

∂ ( S M )

. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension k in terms of the normed homology

$$H_k(M; \mathbb R)$$

H k

( M ; R )

and

$$H_k(DM; \mathbb R)$$

H k

( D M ; R )

, where

$$DM = M\cup _{\partial M} M$$

D M = M

∪

∂ M

M

denotes the double of M. The norms here are the simplicial semi-norms in homology. The more complex the metric on M is, the more numerous the strata of SM and S(DM) are. It turns out that the normed homology spaces form obstructions to the existence of globally k-convex traversally generic metrics on M. We also prove that knowing the geodesic scattering map on M makes it possible to reconstruct the stratified topological type of the space of geodesics, as well as the amenably localized Poincaré duality operators on SM.