Timeline for Explicit metrics
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2020 at 7:30 | comment | added | Bombyx mori | Thank you a lot for the comment. | |
| Jan 7, 2020 at 6:57 | comment | added | Robert Bryant | @Bombyxmori: One cannot isometrically immerse in a $C^2$ manner a compact surface of nonpositive curvature in Euclidean 3-space because the curvature at a point of maximum distance from the origin will have to be positive. A theorem of Efimov (I think) says that you cannot immerse isometrically in a $C^2$ manner into Euclidean 3-space any complete surface whose curvature is bounded above by a negative constant. Thus, the embeddings of the type you are asking about do not exist if you require that they be $C^2$. By a theorem of Nash, $C^1$ isometric embeddings do exist, but they are wild. | |
| Jan 7, 2020 at 6:22 | comment | added | Bombyx mori | Is it possible to make sense of how to embed a hyper-elliptic curve of type $z^2=f(w)$ to the 3D Euclidean space? I can construct a metric of constant negative curvature on the hyper-elliptic curve, but I do not know any explicit embedding that may work. Curiously (according to C.N.Yang) it seems this (Thurston's) problem was considered by Chern in 1930s already. | |
| Jun 6, 2011 at 19:46 | history | answered | Robert Bryant | CC BY-SA 3.0 |