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Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. ==Statement of the theorem== Let (''M'', ''g'') be a connected Riemannian manifold. Then the following statements are equivalent: # The closed and bounded subsets of ''M'' are compact; # ''M'' is a complete metric space; # ''M'' is geodesically complete; that is, for every ''p'' in ''M'', the exponential map exp''p'' is defined on the entire tangent space T''p''''M''. Furthermore, any one of the above implies that given any two points ''p'' and ''q'' in ''M'', there exists a length minimizing geodesic connecting these two points (geodesics are in general critical points for the length functional, and may or may not be minima). 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「Hopf–Rinow theorem」の詳細全文を読む スポンサード リンク
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