Soul theorem: Difference between revisions
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Any [[open Riemannian manifold]] <math>(M,g)</math> with nonnegative [[sectional curvature]] contains a compact [[totally geodesic submanifold]] <math>S</math> (called the '''soul'''), such that <math>M</math> is diffeomorphic to the [[normal bundle]] on <math>S</math>, viz <math>\nu(S)</math>. | Any [[open Riemannian manifold]] <math>(M,g)</math> with nonnegative [[sectional curvature]] contains a compact [[totally geodesic submanifold]] <math>S</math> (called the '''soul'''), such that <math>M</math> is diffeomorphic to the [[normal bundle]] on <math>S</math>, viz <math>\nu(S)</math>. | ||
==References== | |||
* ''Improving the metric in an open manifold with nonnegative curvature'' by Luis Guijarro, ''Proceedings of the American Mathematical Society, Volume 126, Number 5, May 1998, Pages 1541-1545'' | |||
==External links== | |||
* [http://www.ams.org/proc/1998-126-05/S0002-9939-98-04287-7/S0002-9939-98-04287-7.pdf ''Improving the metric in an open manifold with nonnegative curvature''] | |||
Revision as of 01:37, 7 July 2007
Statement
Any open Riemannian manifold with nonnegative sectional curvature contains a compact totally geodesic submanifold (called the soul), such that is diffeomorphic to the normal bundle on , viz .
References
- Improving the metric in an open manifold with nonnegative curvature by Luis Guijarro, Proceedings of the American Mathematical Society, Volume 126, Number 5, May 1998, Pages 1541-1545