The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.
By the Gauss equation and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature.
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.
As mentioned above, Gromoll and Meyer proved that if g has positive sectional curvature then the soul is a point.
[8] 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.