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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Exponential map (Riemannian geometry) ： ウィキペディア英語版 Exponential map (Riemannian geometry) In Riemannian geometry, an exponential map is a map from a subset of a tangent space T''p''''M'' of a Riemannian manifold (or pseudo-Riemannian manifold) ''M'' to ''M'' itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection. == Definition == Let be a differentiable manifold and a point of . An affine connection on allows one to define the notion of a geodesic through the point .〔A source for this section is , which uses the term "linear connection" where we use "affine connection" instead.〕 Let be a tangent vector to the manifold at . Then there is a unique geodesic satisfying with initial tangent vector . The corresponding exponential map is defined by . In general, the exponential map is only ''locally defined'', that is, it only takes a small neighborhood of the origin at , to a neighborhood of in the manifold. This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle. 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「Exponential map (Riemannian geometry)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.