Deformative Magnetic Marked Length Spectrum Rigidity
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914960328622080 |
|---|---|
| author | Reber, James Marshall |
| author_facet | Reber, James Marshall |
| contents | Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{κ_s : M \rightarrow \mathbb{R} \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, κ_s) \ | \ s \in (-ε, ε)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-ε, ε)$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s : M \rightarrow M \ | \ s \in (-ε, ε)\}$ such that $f_s^*(g_s) = g_0$ and $f_s^*(κ_s) = κ_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01865 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformative Magnetic Marked Length Spectrum Rigidity Reber, James Marshall Differential Geometry Dynamical Systems 58J50 (Primary) 37C10 (Secondary) Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{κ_s : M \rightarrow \mathbb{R} \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, κ_s) \ | \ s \in (-ε, ε)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-ε, ε)$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s : M \rightarrow M \ | \ s \in (-ε, ε)\}$ such that $f_s^*(g_s) = g_0$ and $f_s^*(κ_s) = κ_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows. |
| title | Deformative Magnetic Marked Length Spectrum Rigidity |
| topic | Differential Geometry Dynamical Systems 58J50 (Primary) 37C10 (Secondary) |
| url | https://arxiv.org/abs/2211.01865 |