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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 Double tangent bundle ： ウィキペディア英語版 Double tangent bundle In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space ''TM'' of the tangent bundle of a smooth manifold ''M'' .〔J.M.Lee, ''Introduction to Smooth Manifolds'', Springer-Verlag, 2003.〕 A note on notation: in this article, we denote projection maps by their domains, e.g., ''π''''TTM'' : ''TTM'' → ''TM''. Some authors index these maps by their ranges instead, so for them, that map would be written ''π''''TM''. The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle. == Secondary vector bundle structure and canonical flip == Since is a vector bundle on its own right, its tangent bundle has the secondary vector bundle structure where is the push-forward of the canonical projection In the following we denote :$$ \xi = \xi^k\frac\Big|_x\in T_xM, \qquad X = X^k\frac\Big|_x\in T_xM and apply the associated coordinate system :$$ \xi \mapsto (x^1,\ldots,x^n,\xi^1,\ldots,\xi^n) on ''TM''. Then the fibre of the secondary vector bundle structure at ''X''∈''T''''x''''M'' takes the form :$$ (\pi_)^_ *(X) = \Big\\Big|_\xi + Y^k\frac\Big|_\xi \ \Big| \ \xi\in T_xM \ , \ Y^1,\ldots,Y^n\in\R \ \Big\}. The double tangent bundle is a double vector bundle. The canonical flip〔P.Michor. ''Topics in Differential Geometry,'' American Mathematical Society, 2008.〕 is a smooth involution ''j'':''TTM''→''TTM'' that exchanges these vector space structures in the sense that it is a vector bundle isomorphism between and In the associated coordinates on ''TM'' it reads as :$$ j\Big(X^k\frac\Big|_\xi + Y^k\frac\Big|_\xi\Big) = \xi^k\frac\Big|_X + Y^k\frac\Big|_X. The canonical flip has the property that for any ''f'': R2 → ''M'', :$$ \frac = j \circ \frac where ''s'' and ''t'' are coordinates of the standard basis of R 2. Note that both partial derivatives are functions from R2 to ''TTM''. This property can, in fact, be used to give an intrinsic definition of the canonical flip.〔Robert J. Fisher and H. Turner Laquer, Second Order Tangent Vectors in Riemannian Geometry, J. Korean Math. Soc. 36 (1999), No. 5, pp. 959-1008〕 Indeed, there is a submersion ''p'': J20 (R2,M) → ''TTM'' given by :$$ p(())=\frac (0,0) where ''p'' can be defined in the space of two-jets at zero because only depends on ''f'' up to order two at zero. We consider the application: :$$ J: J^2_0(\mathbb^2,M) \to J^2_0(\mathbb^2,M) \quad / \quad J(())=(\circ \alpha ) where α(''s'',''t'')= (''t'',''s''). Then ''J'' is compatible with the projection ''p'' and induces the canonical flip on the quotient ''TTM''. 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「Double tangent bundle」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.