Timeline for Reference request for Poincare-Hopf theorem in a compact submanifold
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2022 at 19:14 | comment | added | Ninpou | Apparently my last comment was redirected to the chat | |
| Sep 6, 2022 at 6:23 | comment | added | Ryan Budney | So are you saying the zeros have cluster points in $S^1 \times [-H,H]$ for some finite value of $H$? | |
| Sep 6, 2022 at 0:49 | comment | added | Ninpou | I was wondering if I can choose a compact submanifold $N, \partial N \subset M$ where V does not vanish on the boundary $\partial N$. Then I would compute the Poincare-Hopf theorem for the submanifold $N, \partial N$. Since I know the behavior of the vector field at infinity, it would be sufficient for me to know the index of $V$ in a compact region $S^1\times[-H, H]$ where H is a larger number. | |
| Sep 6, 2022 at 0:46 | comment | added | Ninpou | The manifold is the infinity cylinder $M=S^1\times \mathbb{R}$. I have a vector field $V$ defined all over $M$. I know, from the context of my problem, that this vector field does not have isolated zeroes for high values of height $H$. I want to compute some topological invariant related to the zeros of $V$. | |
| Sep 5, 2022 at 19:24 | comment | added | Ryan Budney | For non-compact manifolds you have the problem that the Euler characteristic may not be defined. Are you restricting to some class of manifolds where it is defined? | |
| Sep 5, 2022 at 19:03 | comment | added | Ninpou | I am trying to understand how one can adapt the Poincare-Hopf index theorem to a non-compact manifold. In particular, I am considering a vector field that does not vanish at the infinity of my non-compact manifold. The answer of Professor Bill Thurston is exactly what I am looking for. I would like to read a reference to improve my knowledge (I am self studying the Poincare-Hopf theorem and this topic is new to me). | |
| Sep 5, 2022 at 18:22 | comment | added | Ryan Budney | Could you be a little more specific on what you would like details on? This argument about adapting vector fields to triangulations is a standard one. You can likely find it in a few differential topology textbooks, for example I think Guillemin and Pollack has a version of this kind of argument. | |
| S Sep 5, 2022 at 14:58 | review | First questions | |||
| Sep 5, 2022 at 15:10 | |||||
| S Sep 5, 2022 at 14:58 | history | asked | Ninpou | CC BY-SA 4.0 |