Difference between revisions of "Ricci theorem"
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| + | In order that a surface $ S $ | ||
| + | with metric $ d s ^ {2} $ | ||
| + | and [[Gaussian curvature|Gaussian curvature]] $ K \leq 0 $ | ||
| + | be locally isometric to some [[Minimal surface|minimal surface]] $ F $ | ||
| + | it is necessary and sufficient that (at all points where $ K < 0 $) | ||
| + | the metric $ d \widetilde{s} {} ^ {2} = \sqrt {- K } d s ^ {2} $ | ||
| + | be of Gaussian curvature $ \widetilde{K} = 0 $. | ||
There are generalizations [[#References|[1]]], describing Riemannian metrics which arise as metrics of minimal submanifolds in Euclidean spaces of arbitrary dimension. | There are generalizations [[#References|[1]]], describing Riemannian metrics which arise as metrics of minimal submanifolds in Euclidean spaces of arbitrary dimension. | ||
Latest revision as of 08:11, 6 June 2020
In order that a surface $ S $
with metric $ d s ^ {2} $
and Gaussian curvature $ K \leq 0 $
be locally isometric to some minimal surface $ F $
it is necessary and sufficient that (at all points where $ K < 0 $)
the metric $ d \widetilde{s} {} ^ {2} = \sqrt {- K } d s ^ {2} $
be of Gaussian curvature $ \widetilde{K} = 0 $.
There are generalizations [1], describing Riemannian metrics which arise as metrics of minimal submanifolds in Euclidean spaces of arbitrary dimension.
References
| [1] | S.-S. Chern, R. Osserman, "Remarks on the Riemannian metrics of a minimal submanifold" E. Looijenga (ed.) D. Siersma (ed.) F. Takens (ed.) , Geometry Symp. (Utrecht, 1980) , Lect. notes in math. , 894 , Springer (1981) pp. 49–90 |
How to Cite This Entry:
Ricci theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_theorem&oldid=48538
Ricci theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_theorem&oldid=48538
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article