Soul theorem

The soul theorem states that every complete, connected, noncompact Riemannian manifold with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact, totally geodesic submanifold.

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Cheeger–Gromoll soul construction: A construction that produces a compact, totally geodesic soul inside a complete noncompact manifold of nonnegative sectional curvature. It is the geometric procedure behind the theorem’s compact core.
Riemannian manifold: A smooth manifold equipped with a smoothly varying inner product on each tangent space. The theorem concerns the global topology of spaces equipped with this geometric structure.
Euclidean space: The flat Riemannian space Rn with its standard Euclidean metric. Its soul is a point, and its normal bundle is all of Euclidean space.
Soul conjecture: A conjecture that a complete noncompact manifold with nonnegative sectional curvature and positive curvature somewhere has a point soul. It sharpens the theorem by predicting when the compact core must collapse to a point.
Busemann function: A function defined from the asymptotic distance behavior of a geodesic ray in a metric space. Concavity properties of Busemann functions help identify convex regions in the construction.
Geodesic completeness: The property that every geodesic extends for all real values of its affine parameter. Completeness prevents the theorem’s ambient manifold from ending after finite geodesic travel.
Cylinder (geometry): The product of a circle with a line, equipped with the product metric. Its circle is a compact totally geodesic soul, with a line as normal fiber.
Cheeger–Gromoll splitting theorem: A theorem that a complete manifold with nonnegative Ricci curvature and a line splits isometrically as a product. It is a related rigidity result linking nonnegative curvature to global product structure.
Convex exhaustion: An increasing sequence of compact convex subsets whose union is a space. The soul construction obtains its compact core through nested convex sets.
Sectional curvature: A measure of curvature assigned to each two-dimensional tangent plane of a Riemannian manifold. The theorem’s curvature assumption is stated plane by plane through this quantity.