MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows
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 o...
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/132071 |
| _version_ | 1811092844167823360 |
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| author | Katz, Gabriel |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Katz, Gabriel |
| author_sort | Katz, Gabriel |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T15:29:54Z |
| format | Article |
| id | mit-1721.1/132071 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:29:54Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1320712023-03-24T19:22:08Z MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows Katz, Gabriel Massachusetts Institute of Technology. Department of Mathematics 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. 2021-09-20T17:41:47Z 2021-09-20T17:41:47Z 2021-02-24 2021-03-19T13:03:48Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/132071 Qualitative Theory of Dynamical Systems. 2021 Feb 24;20(1):19 en https://doi.org/10.1007/s12346-021-00448-y 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. The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Katz, Gabriel MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title | MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title_full | MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title_fullStr | MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title_full_unstemmed | MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title_short | MIGHT BE DUPLICATE: Gromov’s Amenable Localization and Geodesic Flows |
| title_sort | might be duplicate gromov s amenable localization and geodesic flows |
| url | https://hdl.handle.net/1721.1/132071 |
| work_keys_str_mv | AT katzgabriel mightbeduplicategromovsamenablelocalizationandgeodesicflows |