Timeline for Geodesics on orthogonal matrix
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 10, 2023 at 4:15 | vote | accept | Luis Yanka Annalisc | ||
| Jul 10, 2023 at 0:48 | answer | added | Ramiro Lafuente | timeline score: 2 | |
| Jul 6, 2023 at 2:46 | comment | added | Daniel Asimov | I just learned from Robert Bryant's comments at mathoverflow.net/questions/126421/… that except for n = 4, all bi-invariant metrics on SO(n) are unique up to constant multiple. (The exception is essentially because so(4) = so(3) ⊕ so(3).) | |
| Jul 6, 2023 at 0:16 | comment | added | Daniel Asimov | Is it obvious that the riemannian metric that O(n) inherits from R^(n×n) is the same as (or proportional to) the usual bi-invariant metric on O(n) ? | |
| Jul 5, 2023 at 19:46 | comment | added | Ben McKay | For $n=4$, $O(4)$ is isometric as a Riemannian manfold to two copies of $SO(4)$, which is a 2-1 quotient of $S^3\times S^3$ with the usual metric. I think all of this is in many books on Lie groups. Maybe try Stillwell, Naive Lie Theory, although I don't have it with me at the moment. | |
| Jul 5, 2023 at 19:44 | comment | added | Ben McKay | For $O(3)$, as a Riemannian manifold it consists of two copies of the real projective space, with metric the quotient of the metric of the 3-sphere, so the geodesics, which are exactly the one parameter subgroups, are exactly the real projective lines, the quotients of the great circles. So now you know about $n=0,1,2,3$. | |
| Jul 5, 2023 at 17:14 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Fixed spellng
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| Jul 5, 2023 at 11:31 | comment | added | Luis Yanka Annalisc | @MikhailKatzM Can you tell me the precise page? | |
| Jul 5, 2023 at 11:21 | comment | added | Mikhail Katz | Helgason, Sigurdur. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. | |
| Jul 5, 2023 at 11:15 | history | asked | Luis Yanka Annalisc | CC BY-SA 4.0 |