| 词条 | Sharafutdinov's retraction |
| 释义 |
In mathematics, Sharafutdinov's retraction is a construction that gives a retraction of an open non-negatively curved Riemannian manifold onto its soul. It was first used by Sharafutdinov[1] to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric. Perelman later showed that in this setting, Sharafutdinov's retraction is in fact a submersion, thereby essentially settling the soul conjecture.[2] For open non-negatively curved Alexandrov space, Perelman also showed that there exists a Sharafutdinov retraction from the entire space to the soul. However it is not yet known whether this retraction is submetry or not. References1. ^{{citation|first=V. A.|last=Sharafutdinov|title=Convex sets in a manifold of nonnegative curvature|journal=Mathematical Notes|volume=26|number=1|year=1979 |pages=556–560 |doi=10.1007/BF01140282}} {{DEFAULTSORT:Sharafutdinov's Retraction}}2. ^{{Citation | last1=Perelman | first1=Grigori | author1-link=Grigori Perelman | title=Proof of the soul conjecture of Cheeger and Gromoll | url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214455292 |mr=1285534 | year=1994 | journal=Journal of Differential Geometry | volume=40 | issue=1 | pages=209–212}} 1 : Riemannian geometry |
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