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In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture was formulated by Gromoll and Cheeger in 1972 and proved by Perelman in 1994 with an astonishingly concise proof. The soul theorem states: :If is a complete connected Riemannian manifold with sectional curvature , then there exists a compact totally convex, totally geodesic submanifold such that is diffeomorphic to the normal bundle of . The submanifold is called a soul of . The soul is not uniquely determined by in general, but any two souls of are isometric. This was proven by Sharafutdinov using Sharafutdinov's retraction in 1979. ==Examples== Every compact manifold is its own soul. Indeed, the theorem is often stated only for non-compact manifolds. As a very simple example, take to be Euclidean space . The sectional curvature is , and any point of can serve as a soul of . Now take the paraboloid }, with the metric being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space . Here the sectional curvature is positive everywhere. The origin is a soul of . Not every point of is a soul of , since there may be geodesic loops based at . One can also consider an infinite cylinder }, again with the induced Euclidean metric. The sectional curvature is everywhere. Any "horizontal" circle } with fixed is a soul of . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Soul theorem」の詳細全文を読む スポンサード リンク
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