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

Deformation retract

In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...

 for a given concave function

Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...

 on a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

.

It was first used by Sharafutdinov to show that any two souls

Soul theorem

In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...

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

Isometry (Riemannian geometry)

In the study of Riemannian geometry in mathematics, a local isometry from one Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first...

. Perelman

Grigori Perelman

Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

 later showed that in this setting, Sharafutdinov's retraction is in fact a submersion

Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...

, thereby essentially settling the soul conjecture

Soul theorem

In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...

.