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^[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.

References

1. Sharafutdinov, V. A. (1979), "Convex sets in a manifold of nonnegative curvature", Mathematical Notes 26 (1): 556–560, doi:10.1007/BF01140282
2. Perelman, Grigori (1994), "Proof of the soul conjecture of Cheeger and Gromoll", Journal of Differential Geometry 40 (1): 209–212, MR 1285534