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.
The soul theorem states:
- 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).
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
- Cheeger, Jeff; Gromoll, Detlef (1972), "On the structure of complete manifolds of nonnegative curvature", Annals of Mathematics. Second Series 96: 413–443, MR0309010, ISSN 0003-486X, http://www.jstor.org/stable/1970819
- Gromoll, Detlef; Meyer, Wolfgang (1969), "On complete open manifolds of positive curvature", Annals of Mathematics. Second Series 90: 75–90, MR0247590, ISSN 0003-486X, http://www.jstor.org/stable/1970682
- Perelman, Grigori (1994), "Proof of the soul conjecture of Cheeger and Gromoll", Journal of Differential Geometry 40 (1): 209–212, MR1285534, ISSN 0022-040X, http://projecteuclid.org/getRecord?id=euclid.jdg/1214455292
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