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 \(\partial (SM)\). 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 \(H_k(M; \mathbb R)\) and \(H_k(DM; \mathbb R)\), where \(DM = M\cup _{\partial M} M\) 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.

让中号是一个紧致光滑黎曼ň -manifold与边界。我们将Gromov的适应性定位技术与Poincaré对偶性相结合，以研究球形切线束空间SM上的遍历通用测地流。这样的流程会生成SM的分层，并由丰富的通用组合函数控制。分层反映了流动轨迹与边界\（\ partial（SM）\）相切的方式。具体地，我们得到任何给定的余维的这些流动产生的地层连通分量的数目的低估计ķ在赋范同源性而言\（H_k（M; \ mathbb R）\）和\（H_k（DM; \ mathbb R）\），其中\（DM = M \ cup _ {\ partial M} M \）表示M的两倍。这里的范数是同源性中的简单半范数。M上的度量越复杂，SM和S（DM）的层数就越多。事实证明，在赋范同源性空间形成障碍物全局的存在ķ -凸上traversally通用度量中号。我们还证明了知道M上的测地线散射图使得有可能重建测地线空间的分层拓扑类型，以及在SM上合适的局部Poincaré对偶算子。