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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[1] Sharafutdinov, V. A. (1979), "Convex sets in a manifold of nonnegative curvature", Mathematical Notes, 26 (1): 556–560, doi:10.1007/BF01140282, S2CID 119764156

[2] 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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