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Difference between revisions of "Ricci theorem"

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In order that a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818101.png" /> with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818102.png" /> and [[Gaussian curvature|Gaussian curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818103.png" /> be locally isometric to some [[Minimal surface|minimal surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818104.png" /> it is necessary and sufficient that (at all points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818105.png" />) the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818106.png" /> be of Gaussian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081810/r0818107.png" />.
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In order that a surface  $  S $
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with metric  $  d s  ^ {2} $
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and [[Gaussian curvature|Gaussian curvature]] $  K \leq  0 $
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be locally isometric to some [[Minimal surface|minimal surface]] $  F $
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it is necessary and sufficient that (at all points where $  K < 0 $)  
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the metric $  d \widetilde{s}  {}  ^ {2} = \sqrt {- K }  d s  ^ {2} $
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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=19304
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article