Timeline for Special map from a manifold to GL_n(R)?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 12, 2010 at 2:45 | vote | accept | Zev Chonoles | ||
| Dec 16, 2009 at 1:24 | answer | added | Greg Kuperberg | timeline score: 9 | |
| Dec 15, 2009 at 10:30 | comment | added | Andrew Stacey | Whoops! Okay, ignore that comment. | |
| Dec 15, 2009 at 9:18 | comment | added | Mariano Suárez-Álvarez | @Andrew that works if f is the identity only :P | |
| Dec 15, 2009 at 8:56 | comment | added | Andrew Stacey | If you take M = N, so have a diffeomorphism f : M -> M, then you get a map from M to the "conjugation space" of GL_n(R): mod out by the relation of similarity. This isn't a nice manifold, though. | |
| Dec 15, 2009 at 6:42 | comment | added | Zev Chonoles | Thanks for your explanation, Mariano - makes sense! And thanks for another canonical map Evgeny :) Is there a standard name for a section of hom($TM$,$TN$)? | |
| Dec 15, 2009 at 5:00 | comment | added | Harry Gindi | Pff, real men work with frechet manifolds. | |
| Dec 15, 2009 at 4:30 | comment | added | Evgeny Shinder | Comment to the last part of your question. An n-manifold M embedded into R^N defines canonically a map from M to Grassmannian Gr(n,N): the image of a point p in M is a tangent space at that point as a subspace of R^N. It is an analog of Gauss map for higher dimensional manifolds without orientation and metric. | |
| Dec 15, 2009 at 3:54 | comment | added | Mariano Suárez-Álvarez | You cannot define such a map unless you are able to pick fields on $M$ and on $N$ such that they are a basis at each point of $M$ and $N$ (because the matrix of a linear map depends on a choice of bases). You'd need both $M$ and $N$ to be [parallelizable]{mathworld.wolfram.com/Parallelizable.html). What you can do is construct a bundle $\hom(TM,TN)$ and define your $f$ to be a section of this bundle: this is a standard trick to avoid to pick bases. | |
| Dec 15, 2009 at 3:49 | history | asked | Zev Chonoles | CC BY-SA 2.5 |