the Cheeger-Gromoll soul theorem
/ CHAY-gur GROH-mol /
A complete surface that is nonnegatively curved and open — like an infinite paraboloid or a flat cylinder — seems to stretch off forever, but it always has a compact 'core' it wraps around, and everything else is just the core thickened out by straight directions. The Cheeger-Gromoll soul theorem makes that picture exact in every dimension. It says any complete, noncompact, nonnegatively curved manifold deformation-retracts onto a compact, totally geodesic submanifold — its soul — and the whole manifold is nothing more than the normal bundle of that soul.
Precisely: let M be a complete, noncompact Riemannian manifold with sectional curvature K >= 0. Then there exists a compact, totally geodesic, totally convex submanifold S (the soul) such that M is diffeomorphic to the total space of the normal bundle of S in M. In particular M deformation-retracts onto S, so M and S have the same homotopy type, and the noncompactness is entirely carried by the fibre directions. The construction is beautiful: starting from any point, exhaust M by a nested family of totally convex sets built from the distance function (Busemann functions of rays), and shrink them down; what remains and cannot shrink further is the soul. A totally convex set is one containing every geodesic between any two of its points; total geodesy of S means S is itself a geodesic submanifold (geodesics of S are geodesics of M).
The soul theorem is the structural backbone for noncompact nonnegative curvature, reducing the topology of an open manifold to that of a compact piece. Its famous companion is the Cheeger-Gromoll-Perelman soul conjecture (proved by Perelman in 1994): if the curvature is strictly positive at even a single point, the soul is a single point, so M is diffeomorphic to R^n. Honest caveats. First, nonnegative SECTIONAL curvature is required — a Ricci bound is genuinely weaker and does not give a soul. Second, 'totally geodesic' and 'totally convex' are stronger than 'minimal' or 'convex'; the soul is not merely a minimal submanifold. Third, the soul need not be unique as a set, though all souls are isometric, and the diffeomorphism type of M is what is pinned down.
The flat infinite cylinder S^1 x R has K = 0 >= 0, is complete and noncompact; its soul is the central circle S^1 x {0}, a compact totally geodesic submanifold, and the cylinder is exactly the (trivial) normal line bundle over that circle — it deformation-retracts onto the circle. The flat plane R^2 has K = 0 with a soul that is a single point (any point), and R^2 is the normal bundle of a point, namely R^2 itself. The paraboloid of revolution has K > 0 strictly everywhere, so by Perelman's resolution its soul is a point and it is diffeomorphic to R^2 — which it visibly is.
Cylinder: soul = central circle; plane and paraboloid: soul = a point. M is the normal bundle of its soul.
The soul theorem needs nonnegative SECTIONAL curvature and noncompactness; it says nothing about compact manifolds. And the soul is totally geodesic and totally convex, far stronger than minimal — do not weaken these to mere convexity.