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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 Gauss–Codazzi equation ： ウィキペディア英語版 Gauss–Codazzi equations In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds. In the classical differential geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. The first equation, sometimes called the Gauss equation, relates the ''intrinsic curvature'' (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form. This equation is the basis for Gauss's theorema egregium.〔.〕 The second equation, sometimes called the Codazzi–Mainardi equation, is a structural condition on the second derivatives of the Gauss map. It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result,〔.〕 although it was discovered earlier by .〔.〕 It incorporates the ''extrinsic curvature'' (or mean curvature) of the surface. The equations show that the components of the second fundamental form and its derivatives along the surface completely classify the surface up to a Euclidean transformation, a theorem of Ossian Bonnet.〔.〕 ==Formal statement== Let i : ''M'' ⊂ ''P'' be an ''n''-dimensional embedded submanifold of a Riemannian manifold ''P'' of dimension ''n''+''p''. There is a natural inclusion of the tangent bundle of ''M'' into that of ''P'' by the pushforward, and the cokernel is the normal bundle of ''M'': :$0\backslash rightarrow\; T\_xM\; \backslash rightarrow\; T\_xP|\_M\; \backslash rightarrow\; T\_x^\backslash perp\; M\backslash rightarrow\; 0.$ The metric splits this short exact sequence, and so :$TP|\_M\; =\; TM\backslash oplus\; T^\backslash perp\; M.$ Relative to this splitting, the Levi-Civita connection ∇′ of ''P'' decomposes into tangential and normal components. For each ''X'' ∈ T''M'' and vector field ''Y'' on ''M'', :$\backslash nabla\text{'}\_X\; Y\; =\; \backslash top(\backslash nabla\text{'}\_X\; Y)\; +\; \backslash bot(\backslash nabla\text{'}\_X\; Y).$ Let :$\backslash nabla\_X\; Y\; =\; \backslash top(\backslash nabla\text{'}\_X\; Y),\backslash quad\; \backslash alpha(X,Y)\; =\; \backslash bot(\backslash nabla\text{'}\_X\; Y).$ Gauss' formula〔Terminology from Spivak, Volume III.〕 now asserts that ∇X is the Levi-Civita connection for ''M'', and α is a ''symmetric'' vector-valued form with values in the normal bundle. It is often referred to as the second fundamental form. An immediate corollary is the Gauss equation. For ''X'', ''Y'', ''Z'', ''W'' ∈ T''M'', :$\backslash langle\; R\text{'}(X,Y)Z,\; W\backslash rangle\; =\; \backslash langle\; R(X,Y)Z,\; W\backslash rangle\; +\; \backslash langle\; \backslash alpha(X,Z),\; \backslash alpha(Y,W)\backslash rangle\; -\backslash langle\; \backslash alpha(Y,Z),\; \backslash alpha(X,W)\backslash rangle$ where ''R''′ is the Riemann curvature tensor of ''P'' and ''R'' is that of ''M''. The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let ''X'' ∈ T''M'' and ξ a normal vector field. Then decompose the ambient covariant derivative of ξ along ''X'' into tangential and normal components: :$\backslash nabla\text{'}\_X\backslash xi=\backslash top\; (\backslash nabla\text{'}\_X\backslash xi)\; +\; \backslash bot(\backslash nabla\text{'}\_X\backslash xi)\; =\; -A\_\backslash xi(X)\; +\; D\_X(\backslash xi).$ Then # ''Weingarten's equation'': $\backslash langle\; A\_\backslash xi\; X,\; Y\backslash rangle\; =\; \backslash langle\; \backslash alpha(X,Y),\; \backslash xi\backslash rangle$ # ''D''X is a metric connection in the normal bundle. There are thus a pair of connections: ∇, defined on the tangent bundle of ''M''; and ''D'', defined on the normal bundle of ''M''. These combine to form a connection on any tensor product of copies of T''M'' and T⊥''M''. In particular, they defined the covariant derivative of α: :$(\backslash tilde\_X\; \backslash alpha)(Y,Z)\; =\; D\_X\backslash left(\backslash alpha(Y,Z)\backslash right)\; -\; \backslash alpha(\backslash nabla\_X\; Y,Z)\; -\; \backslash alpha(Y,\backslash nabla\_X\; Z).$ The Codazzi–Mainardi equation is :$\backslash bot\backslash left(R\text{'}(X,Y)Z\backslash right)\; =\; (\backslash tilde\_X\backslash alpha)(Y,Z)\; -\; (\backslash tilde\_Y\backslash alpha)(X,Z).$ Since every immersion is, in particular, a local embedding, the above formulas also hold for immersions. 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–Codazzi equations」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.