In mathematics, the soul theorem is the following theorem of Riemannian geometry: Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...

If (M,g) is a complete non-compact Riemannian manifold with sectional curvature K ≥ 0, then (M,g) has 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, but any two souls are isometric. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... This is a glossary of terms specific to differential geometry and differential topology. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

Cheeger and Gromoll (1972) proved the theorem by generalizing a result in Gromoll and Meyer (1969).

Soul conjecture

Cheeger and Gromoll (1972) also set out the following conjecture:

Suppose M is complete and noncompact with sectional curvature K ≥ 0, with K > 0 holding at some point. Then the soul of M has to be a point; equivalently M is diffeomorphic to .

Perelman (1994) verified the conjecture with an astonishingly concise proof.

References

• Jeff Cheeger and Gromoll, Detlef (1972) "On the structure of complete manifolds of nonnegative curvature," Ann. of Math. 96: 413-43.
• Gromoll, Detlef, and Meyer, Wolfgang (1969) "On complete open manifolds of positive curvature," Ann. of Math. 90: 75-90.
• Grigory Perelman (1994) "Proof of the soul conjecture of Cheeger and Gromoll," J. Differential Geom. 40: 209-12.