Timeline for submanifold of a hyperbolic manifold
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 5, 2019 at 14:58 | vote | accept | Nick L | ||
| Apr 5, 2019 at 14:39 | comment | added | Deane Yang | I know very little, beyond the definition, about hyperbolic manifolds, but I do know a lot about isometric embeddings. I agree with Misha and Andy that isometric embeddings without additional assumptions are unlikely to be useful. There are too many degrees of freedom in the embedding, so it's really hard to extract useful geometric information from the embedding. In particular, the key local invariant is the second fundamental form, and in codimensions 2 or more, it is a family of symmetric tensors satisfying the Gauss equations, which is a real algebraic variety and hard to study. | |
| Apr 5, 2019 at 14:23 | answer | added | Misha | timeline score: 10 | |
| Apr 5, 2019 at 3:48 | comment | added | Andy Putman | I think that most of what is known about this kind of question is summarized in the introduction to the following paper: arxiv.org/abs/1802.04619 | |
| Apr 5, 2019 at 3:39 | comment | added | Nick L | Thanks Misha! Please can give a reference for the statement about isometric embeddings in dimension $\geq 7$? Also, do you know what happens in dimensions $4,5$ and $6$? | |
| Apr 5, 2019 at 3:37 | comment | added | Misha | I suspect you want a subsurface to be totally geodesic; otherwise it is quite useless (you can embed isometrically every surface in every hyperbolic manifold of dimension $\ge 7$ or so). On the positive side, by Kahn-Markovic, there are always almost totally geodesic compact immersed subsurfaces (and their fundamental groups are embedded). But if you want totally geodesic surfaces, there are no good conditions. | |
| Apr 5, 2019 at 2:29 | history | asked | Nick L | CC BY-SA 4.0 |