In mathematics, Sharafutdinov's retraction is a construction that gives a

retraction Retraction or retract(ed) may refer to: Academia * Retraction in academic publishing, withdrawals of previously published academic journal articles Mathematics * Retraction (category theory) * Retract (group theory) * Retraction (topology) Huma ...

of an open non-negatively curved

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

onto its soul. It was first used by Sharafutdinov to show that any two

souls In many religious and philosophical traditions, there is a belief that a soul is "the immaterial aspect or essence of a human being". Etymology The Modern English noun ''soul'' is derived from Old English ''sāwol, sāwel''. The earliest attes ...

of a complete Riemannian manifold with non-negative sectional curvature are

isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...

. Perelman later showed that in this setting, Sharafutdinov's retraction is in fact a submersion, thereby essentially settling the soul conjecture. 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 This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provid ...

or not.

References

Riemannian geometry {{Riemannian-geometry-stub