Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane
| dc.creator | Brander, David | |
| dc.date | 2007-10-02 | |
| dc.date | 2008-01-30 | |
| dc.date.accessioned | 2026-07-07T08:56:59Z | |
| dc.date.available | 2026-07-07T08:56:59Z | |
| dc.description | We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (non-compact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space $H^n$ into a Euclidean space $E^{n+k}$, $n \geq 2$, and the problem of Lagrangian isometric immersions of $H^n$ into $\cc^n$, $n \geq 2$. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not. | |
| dc.description | 8 Pages. Comments and references added | |
| dc.identifier | https://arxiv.org/abs/0710.0507 | |
| dc.identifier | http://arxiv.org/abs/0710.0507 | |
| dc.identifier | Electron. Res. Announc. Math. Sci. 15 (2008), 8 - 16 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146807 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53C42 (Primary); 37K25, 53C35 (Secondary) | |
| dc.title | Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane | |
| dc.type | text |