Timeline for Unstable manifolds of a Morse function give a CW complex
Current License: CC BY-SA 4.0
11 events
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| Nov 24, 2019 at 23:07 | history | edited | alesia | CC BY-SA 4.0 |
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| Nov 24, 2019 at 23:06 | comment | added | alesia | numdam.org/article/ASNSP_1963_3_17_1-2_97_0.pdf (the fact that neighborhoods of critical points are unaffected is explicit at page 109) | |
| Nov 24, 2019 at 22:18 | comment | added | Dmitri Panov | Thanks Alesia. Do you know some good reference for your first phrase: "By Kupka-Smale theorem, a Morse function f on a manifold with a Riemmannian metric m can be perturbed to become Morse-Smale, by genericity." And for your last phrase: "Fortunately the proof of the genericity of Morse-Smale condition produces a perturbation that does not affect those neighborhoods" | |
| Nov 24, 2019 at 19:24 | comment | added | alesia | yes I hadn't addressed that part. Edited the answer to include it. | |
| Nov 24, 2019 at 19:23 | history | edited | alesia | CC BY-SA 4.0 |
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| Nov 24, 2019 at 18:51 | comment | added | Dmitri Panov | I see! This is not quite what I was asking for, the condition that Laudenbach imposes in his paper is that the gradient vector field has a very simple shape : $-x_1\frac{\partial}{\partial x_1}-\ldots -x_i\frac{\partial}{\partial x_i}+x_{i+1}\frac{\partial}{\partial x_{i+1}} +...$ close to each critical point. See proof of his Propositon 2. But thanks anyway, I'll try to understand why what your answer proves what you say. And I wanted to see why such a metric exists behaving in this nice way close to critical points | |
| Nov 24, 2019 at 18:45 | comment | added | alesia | I am showing that there exists a metric for which stable and unstable manifolds of $f$ are transverse, assuming $f$ is a Morse function. | |
| Nov 24, 2019 at 18:43 | history | edited | alesia | CC BY-SA 4.0 |
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| Nov 24, 2019 at 18:41 | comment | added | Dmitri Panov | Alesia, thanks. I deleted my comments, so that there are not too many. My next question is the following: What question you are answering? What are you proving? Could you please say it in the very beginning of your answer? (unfortunately I can not understand this so far) | |
| Nov 24, 2019 at 18:29 | history | edited | alesia | CC BY-SA 4.0 |
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| Nov 24, 2019 at 16:51 | history | answered | alesia | CC BY-SA 4.0 |