翻訳と辞書 Words near each other ・ Gausian dynasty ・ Gauslaa ・ Gauss (crater) ・ Gauss (disambiguation) ・ Gauss (ship) ・ GAUSS (software) ・ Gauss (surname) ・ Gauss (unit) ・ Gauss Centre for Supercomputing ・ Gauss circle problem ・ Gauss expedition ・ Gauss Glacier ・ Gauss iterated map ・ Gauss Lectureship ・ Gauss lens ・ Gauss map ・ Gauss map (disambiguation) ・ Gauss Moutinho Cordeiro ・ Gauss Peninsula ・ Gauss pseudospectral method ・ Gauss Research Laboratory ・ Gauss Speaker Company ・ GAUSS Srl ・ Gauss sum ・ Gauss Tower ・ Gauss' Method ・ Gauss's constant ・ Gauss's continued fraction ・ Gauss's diary ・ Gauss's inequality Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gauss map ： ウィキペディア英語版 Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p'', namely the normal vector to ''X'' at ''p''. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator. Gauss first wrote a draft on the topic in 1825 and published in 1827. There is also a Gauss map for a link, which computes linking number. ==Generalizations== The Gauss map can be defined for hypersurfaces in R''n'' as a map from a hypersurface to the unit sphere ''S''''n'' − 1 ∈ R''n''. For a general oriented ''k''-submanifold of R''n'' the Gauss map can be also be defined, and its target space is the ''oriented'' Grassmannian $\backslash tilde\_$, i.e. the set of all oriented ''k''-planes in R''n''. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented ''normal'' subspace; these are equivalent as $\backslash tilde\_\; \backslash cong\; \backslash tilde\_$ via orthogonal complement. In Euclidean 3-space, this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as $\backslash tilde\_\; \backslash cong\; S^$), hence this is consistent with the definition above. Finally, the notion of Gauss map can be generalized to an oriented submanifold ''X'' of dimension ''k'' in an oriented ambient Riemannian manifold ''M'' of dimension ''n''. In that case, the Gauss map then goes from ''X'' to the set of tangent ''k''-planes in the tangent bundle ''TM''. The target space for the Gauss map ''N'' is a Grassmann bundle built on the tangent bundle ''TM''. In the case where $M=\backslash mathbf^n$, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition. 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「Gauss map」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.