Timeline for Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2019 at 9:11 | comment | added | Călin | The reason I used "isomorphism" is because that's what the author of the first reference uses and I wasn't sure myself what it means. After I found the 2nd and 3rd references, it was clear that there is an isometry for the case $n=2$, but I was still wondering what he means for $n > 2$. | |
| May 28, 2019 at 16:58 | comment | added | Robert Bryant | @CălinCruceru: What's true is that both spaces are diffeomorphic to $\mathbb{R}^p$ as smooth manifolds, but that is a very weak statement (and essentially obvious to boot). When you asked about 'isomorphism', I assumed that you were asking about 'isometry'. Because they are both spaces of non-positive sectional curvature (and simply-connected) there is a 'canonical' diffeomorphism between them (once base points have been chosen in each) got by comparing their exponential maps at their respective base points, but, when $n>2$, that map is very far from being an isometry. | |
| May 28, 2019 at 14:20 | comment | added | Călin | I see. Thanks a lot, Robert. It's funny that the author states the property in a pretty generic way in several places, though. | |
| May 28, 2019 at 14:13 | vote | accept | Călin | ||
| May 28, 2019 at 13:59 | history | answered | Robert Bryant | CC BY-SA 4.0 |