![]() Very simple examples, as below, show that the soul is not uniquely determined by ( M, g) in general. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that M is diffeomorphic to Euclidean space. By the Gauss equation and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Such a submanifold is called a soul of ( M, g). The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.Ĭheeger and Gromoll's soul theorem states: If ( M, g) is a complete connected Riemannian manifold with nonnegative sectional curvature, then there exists a closed totally convex, totally geodesic embedded submanifold whose normal bundle is diffeomorphic to M. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. 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. ![]() Complete manifolds of non-negative sectional curvature largely reduce to the compact case ![]()
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