1994/01/01 by G. Perelman · 3 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Conjecture #Mathematics #Mathematics and Applications #Philosophy #Point processes and geometric inequalities #Pure mathematics #Soul #Theology
paper · pdf · doi:10.4310/jdg/1214455292
openalex publication_date 1994/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/18
In this note we consider complete noncompact Riemannian manifolds M of nonnegative sectional curvature. The structure of such manifolds was discovered by Cheeger and Gromoll [2]: M contains a (not necessarily unique) totally convex and totally geodesic submanifold S without boundary, 0 < dimS < dimM, such that M is diffeomorphic to the total space of the normal bundle of *S in . (S is called a soul of M.) In particular, if S is a single point, then M is diffeomorphic to a Euclidean space. This is the case if all sectional curvatures of M are positive, according to an earlier result of Gromoll and Meyer Cheeger and Gromoll conjectured that the same conclusion can be obtained under the weaker assumption that M contains a point where all sectional curvatures are positive. A contrapositive version of this conjecture expresses certain rigidity of manifolds with souls of positive dimension. It was verified in Recently Marenich [4] published an argument for analytic manifolds without dimensional restrictions. (We were unable to get through that argument, containing over 50 pages of computations.)