In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the soul theorem is a theorem of Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

 that largely reduces the study of complete manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s of non-negative sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 to that of the compact case. Cheeger

Jeff Cheeger

Jeff Cheeger , is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and its applications to topology and analysis.-Biography:He graduated from Harvard University with a B.A....

 and Gromoll

Detlef Gromoll

Detlef Gromoll was a mathematician who studied distortions of shapes in three or more dimensions. In his soul theorem, published in 1972, he and Jeff Cheeger studied the properties of surfaces that have flat regions or curves like the outside of a sphere but not regions shaped liked saddles...

 proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer. The related soul conjecture was formulated by Gromoll and Cheeger in 1972 and proved by 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...

 in 1994 with an astonishingly concise proof.

Soul theorem

The soul theorem states:

If (M, g) is a complete

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 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...

 with sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 K ≥ 0, then (M, g) has a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 totally convex, totally geodesic submanifold S such that M is diffeomorphic

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

 to the normal bundle of S.

The submanifold S is called a soul of (M, g).

The soul is in general not uniquely determined by (M, g), but any two souls of (M, g) 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...

. This was proven by Sharafutdinov using Sharafutdinov's retraction

Sharafutdinov's retraction

In mathematics, Sharafutdinov's retraction is a construction that gives a retraction for a given concave function on a Riemannian manifold.It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric...

 in 1979.

Examples

Every compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 manifold is its own soul. Indeed, the theorem is often stated only for non-compact manifolds.

As a very simple example, take M to be Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 Rⁿ. The sectional curvature is 0, and any point of M can serve as a soul of M.

Now take the paraboloid

Paraboloid

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

 M = {(x,y,z) : z = x² + y²}, with the metric g being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space R³. Here the sectional curvature is positive everywhere. The origin (0,0,0) is a soul of M. Not every point x of M is a soul of M, since there may be geodesic loops based at x.

One can also consider an infinite cylinder

Cylinder (geometry)

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

 M = {(x,y,z) : x² + y² = 1}, again with the induced Euclidean metric. The sectional curvature is 0 everywhere. Any "horizontal" circle {(x,y,z) : x² + y² = 1} with fixed z is a soul of M.

Soul conjecture

Cheeger and Gromoll's soul conjecture states:

Suppose (M, g) is complete, connected and non-compact with sectional curvature K ≥ 0, and there exists a point in M where the sectional curvature (in all sectional directions) is strictly positive. Then the soul of M is a point; equivalently M is diffeomorphic to Rⁿ.

Perelman proved this statement by establishing that in the general case K ≥ 0, Sharafutdinov's retraction

Sharafutdinov's retraction

In mathematics, Sharafutdinov's retraction is a construction that gives a retraction for a given concave function on a Riemannian manifold.It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric...

 P : M → S is 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...

.