![]() ![]() ![]() Īs mentioned above, Gromoll and Meyer proved that if g has positive sectional curvature then the soul is a point. Non-horizontal cross sections of the cylinder are not souls since they are neither totally convex nor totally geodesic. Now take the paraboloid M = with fixed z is a soul of M.The sectional curvature is 0 everywhere, and any point of M can serve as a soul of M. As a very simple example, take M to be Euclidean space R n. ![]() For this reason, the theorem is often stated only for non-compact manifolds.
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