Timeline for Invariance group of Morse charts
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2017 at 17:58 | comment | added | Will Sawin | @AliTaghavi Given that it's been almost 5 years since the question was first asked, if you're interested in that question it might make sense to ask it again. However I think the answer is almost certainly no, as $\phi \circ f= \phi$ can be written as infinitely many equations and hence the set where it is satisfied should have infinite codimension. | |
| Jun 19, 2017 at 17:27 | comment | added | Ali Taghavi | @Matthias For a compact manifold $M$ is there a finite dimensional group $G$ of diffeomorphisms of $M$ such that for every Morse function $\phi :M \to \mathbb{R}$ there is a non trivial element $f\in G$ such that $\phi \circ f=\phi$? I admit that this a completely different question but is motivated by your question. | |
| Jun 19, 2017 at 15:25 | comment | added | Ali Taghavi | @Matthias I think that the group of Local diffeomorphism around the origin can be very Large. But what about the following modification: | |
| Jun 19, 2017 at 15:18 | comment | added | Ali Taghavi | @WillSawin a precise example of convergence of the power seris:Assume that $\gamma$ is an analytic curve in $O(n)$ then define the map G with $G(X)= \gamma({|X|^2})X $ is an analytic diffeomorphism satisfies $|G(X)|=|X|$. | |
| Oct 19, 2012 at 0:49 | comment | added | Will Sawin | But it's not clear at all to me what should be the boundary conditions near $0$. | |
| Oct 19, 2012 at 0:47 | comment | added | Will Sawin | If you take the derivatives at the origin you get three power series that formally satisfy the equation $f^2+g^2+h^2=x^2+y^2+z^2$. There are lots of power series that formally satisfy this equation, as you can see by building it up step by step: degree $1$ terms, then degree $2$, then degree $3$, etc. It seems to me that many of them should have positive radius of convergence but I haven't checked. If some of them do have positive radius of convergence that gets you a lot of germs that aren't $O(n)$. | |
| Oct 18, 2012 at 3:20 | comment | added | macbeth | (2.) Yes, we're interested in the diffeomorphisms satisfying $\varphi \circ f = \varphi$. Such a diffeo $f$ has the property that, on each level set $\{x:|x|^2=r\}$, $f$ restricts to a diffeo of the level set. Moreover the possible $f$ are basically characterized by this property. Think of $f$ as a (germ of a) 1-parameter family of diffeomorphisms of the $(n-1)$-sphere, modulo some boundary conditions (tending to the identity near 0) to ensure smoothness at $p$. | |
| Oct 18, 2012 at 3:19 | comment | added | macbeth | Hi Will (and Kofi). (1.) It seems to me that one can work with the set of germs of charts near p, and the group of germs of diffeomorphisms fixing p. Then the action is well-defined and free and transitive. | |
| Oct 17, 2012 at 16:26 | history | answered | Will Sawin | CC BY-SA 3.0 |