Sharafutdinov's retraction
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 to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric.[1] 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.
References
- ^ Sharafutdinov, V. A. (1979), "Convex sets in a manifold of nonnegative curvature", Mathematical Notes, 26 (1): 556–560, doi:10.1007/BF01140282, S2CID 119764156
- ^ Perelman, Grigori (1994), "Proof of the soul conjecture of Cheeger and Gromoll", Journal of Differential Geometry, 40 (1): 209–212, doi:10.4310/jdg/1214455292, MR 1285534
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