Timeline for What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?
Current License: CC BY-SA 3.0
9 events
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| Dec 2, 2012 at 14:11 | comment | added | J. GE | @Robert, your two answers are really helpful. Thanks again. | |
| Dec 2, 2012 at 1:12 | comment | added | Robert Bryant | @GB: Your second question is more philosophical in nature, and I'm not sure that I have a helpful answer. For me (but, apparently, not for everyone), working out a lot of explicit examples when I'm learning the basics of a subject seems to necessary for developing intuition. It also makes it easier to present those examples concretely without having to rely on a lot of theoretical machinery. Hiding the theoretical machinery is a little misleading, though, because it is a critical part of organizing the plethora of examples one examines into a working understanding. | |
| Dec 1, 2012 at 23:20 | comment | added | Robert Bryant | @GB: I'm sure that it's written in the literature in various places, usually on the way to working out some other property. By the way, the above story generalizes in a straightforward way to all of the compact rank one symmetric spaces, so you probably will find some version of it in Besse's book "Manifolds all of whose geodesics are closed". More generally, the lowest nontrivial eigenspace on any compact irreducible Riemannian symmetric space $M=G/K$ provides a $G$-equivariant isometric embedding of $M$ into an irreducible representation space of $G$ that often has interesting properties. | |
| Dec 1, 2012 at 21:09 | comment | added | J. GE | @Robert, Thank you so much for the detailed answer. I am just wondering about two things. 1) Is this point of view written in the literature? 2) If not, then as a students, what is a good way to get these interesting and geometric insights of thinking analysis properties? | |
| Dec 1, 2012 at 20:52 | vote | accept | J. GE | ||
| Nov 23, 2012 at 14:12 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added a clarifying remark and a tie to symplectic geometry
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| Nov 22, 2012 at 17:23 | comment | added | Renato G. Bettiol | @Robert: Thanks for such a nice answer! I've learnt a lot from it that I didn't know about! :) | |
| Nov 22, 2012 at 15:15 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed a typo and added a comment
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| Nov 22, 2012 at 14:38 | history | answered | Robert Bryant | CC BY-SA 3.0 |