MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows
Name
12346_2021_448_ReferencePDF.pdf
Size
233.64 KB
Format
Adobe PDF
Checksum (MD5)
8c5178e3e0bd7e03d2173161618399e2
Author(s)
Katz, Gabriel
Date Issued
February 24, 2021
Publisher
Springer International Publishing
Citation
Qualitative Theory of Dynamical Systems. 2021 Feb 24;20(1):19
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s12346-021-00448-y
