Soul theorem
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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 in Gromoll and Meyer (1969).
[edit] 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.
[edit] 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.
