Soul theorem

Soul theorem

In mathematics, the soul theorem is the following theorem of Riemannian geometry:

: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.

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

oul 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 {mathbb R}^n.

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

References

*Jeff Cheeger and Detlef Gromoll (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. MathSciNet|id=1285534


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