Hilbert's theorem
From Diffgeom
Template:Curvature result for surfaces
Template:Relating curvature with embedding
This result is about manifolds in dimension:2
This article or section of article is sourced from:Wikipedia
Statement
There exists no complete regular surface of constant negative Gaussian curvature, embedded in .
Facts
Note that there could exist regular surfaces of constant negative curvature, but that such regular surfaces must have boundary points or cusp points at which the curvature no longer remains negative.
ALso Hilbert's theorem is of relevance only for orientable surfaces because non-orientable surfaces anyway cannot be embedded in .
Importance
References
- Über Flächen von konstanter Krümmung by David Hilbert, Trans. Amer. Math. Soc. 2 (1901), 87-99