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 Wolfgang Meyer. The related **soul conjecture** was formulated by Gromoll and Cheeger in 1972 and proved by Perelman in 1994 with an astonishingly concise proof.

The **soul theorem** states:

If

(*M*, *g*) is a complete connected Riemannian manifold with sectional curvature

*K* ≥ 0, then there exists a compact totally convex, totally geodesic submanifold

*S* such that

*M* is diffeomorphic to the normal bundle of

*S*.

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

The soul is not uniquely determined by (*M*, *g*) in general, but any two souls of (*M*, *g*) are isometric. This was proven by Sharafutdinov using Sharafutdinov's retraction in 1979.

Every compact 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 **R**^{n}. The sectional curvature is 0, and any point of *M* can serve as a soul of *M*.

Now take the paraboloid *M* = {(*x*, *y*, *z*) : *z* = *x*^{2} + *y*^{2}}, with the metric *g* being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space **R**^{3}. 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 *M* = {(*x*, *y*, *z*) : *x*^{2} + *y*^{2} = 1}, again with the induced Euclidean metric. The sectional curvature is 0 everywhere. Any "horizontal" circle {(*x*, *y*, *z*) : *x*^{2} + *y*^{2} = 1} with fixed *z* is a soul of *M*.

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

**R**^{n}.

Grigori Perelman proved this statement by establishing that in the general case *K* ≥ 0, Sharafutdinov's retraction *P : M → S* is a submersion. Cao and Shaw later provided a different proof that avoids Perelman's flat strip theorem.