of the cylinder are not souls since they are neither totally convex nor totally geodesic. Soul conjecture As mentioned above, Gromoll and Meyer proved that if has positive sectional curvature then the soul is a point. Cheeger and Gromoll conjectured that this would hold even if had nonnegative sectional curvature, with positivity only required of all sectional curvatures at a single point. This soul conjecture was proved by Grigori Perelman, who established the more powerful fact that Sharafutdinov's retraction is a Riemannian submersion, and even a submetry. References Sources. Theorems in Riemannian geometry