en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

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WP:CHECKWIKI error fixes using AWB (10093)

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			In [[mathematics]], particularly [[differential topology]], the '''secondary vector bundle structure'''
			refers to the natural [[vector bundle]] structure (''TE'',''p''<sub></sub>,''TM'') on the total space ''TE'' of the [[tangent bundle]] of a smooth vector bundle (''E'',''p'',''M''), induced by the push-forward ''p''<sub></sub>:''TE''→''TM'' of the original projection map ''p'':''E''→''M''.

			In the special case (''E'',''p'',''M'')=(''TM'',π<sub>''TM''</sub>,''M''), where ''TE''=''TTM'' is the [[double tangent bundle]], the secondary vector bundle (''TTM'',(π<sub>''TM''</sub>)<sub>*</sub>,''TM'') is isomorphic to the [[tangent bundle]]
			(''TTM'',π<sub>''TTM''</sub>,''TM'') of ''TM'' through the [[double tangent bundle\|canonical flip]].

	I'm a ~~48 years old~~ and ~~work at~~ the ~~university~~ (~~Biochemistry~~).<br>~~In my free time I learn English~~. I'~~ve been twicethere~~ and ~~look forward~~ to ~~go there anytime soon~~. ~~I love to read~~, ~~preferably on my kindle~~. ~~I really love to watch~~ 2 ~~Broke Girls~~ and ~~Doctor Who~~ as ~~well as documentaries about anything astronomical. I like Rock climbing~~.<br><br>~~Here~~ is ~~my weblog~~ :: [~~http:~~//~~www~~.~~ofclan~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~88939 Size guaide mountain bike sizing~~.]		== Construction of the secondary vector bundle structure ==

			Let (''E'',''p'',''M'') be a smooth vector bundle of rank ''N''. Then the preimage (''p''<sub></sub>)<sup>-1</sup>(X)⊂''TE'' of any tangent vector ''X''∈''TM'' in the push-forward ''p''<sub></sub>:''TE''→''TM'' of the canonical projection ''p'':''E''→''M'' is a smooth submanifold of dimension 2''N'', and it becomes a vector space with the push-forwards
			:<math>
			+_:T(E\times E)\to TE \quad , \quad \lambda_:TE\to TE
			</math>
			of the original addition and scalar multiplication
			:<math>
			+:E\times E\to E \qquad , \qquad \lambda:E\to E</math>
			as its vector space operations. The triple (''TE'',''p''<sub>*</sub>,''TM'') becomes a smooth vector bundle with these vector space operations on its fibres.

			=== Proof ===

			Let (''U'',φ) be a local coordinate system on the base manifold ''M'' with φ(''x'')=(''x''<sup>1</sup>,...,''x''<sup>n</sup>) and let
			:<math>
			\psi:W \to \varphi(U)\times \mathbb R^N \quad ; \quad \psi(v^k e_k\|_x) := (x^1,\ldots,x^n,v^1,\ldots,v^N)
			</math>
			be a coordinate system on ''E'' adapted to it. Then
			:<math>
			p_*\Big(X^k\frac{\partial}{\partial x^k}\Big\|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_v\Big) = X^k\frac{\partial}{\partial x^k}\Big\|_{p(v)},
			</math>
			so the fiber of the secondary vector bundle structure at ''X''∈''T''<sub>''x''</sub>''M'' is of the form
			:<math>
			p^{-1}_*(X) = \Big\{ \ X^k\frac{\partial}{\partial x^k}\Big\|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_v
			\ \Big\| \ v\in E_x \ , \ Y^1,\ldots,Y^N\in\R \ \Big\}.
			</math>
			Now it turns out that
			:<math>
			\chi\Big(X^k\frac{\partial}{\partial x^k}\Big\|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_v\Big) = \Big(X^k\frac{\partial}{\partial x^k}\Big\|_{p(v)}, (v^1,\ldots,v^N,Y^1,\ldots,Y^N) \Big)
			</math>
			gives a local trivialization χ:''TW''→''TU''×'''R'''<sup>2''N''</sup> for (''TE'',''p''<sub>*</sub>,''TM''), and the push-forwards of the original vector space operations read in the adapted coordinates as
			:<math>
			\Big(X^k\frac{\partial}{\partial x^k}\Big\|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_v\Big)

			+_*
			\Big(X^k\frac{\partial}{\partial x^k}\Big\|_w + Z^\ell\frac{\partial}{\partial v^\ell}\Big\|_w\Big)

			=
			X^k\frac{\partial}{\partial x^k}\Big\|_{v+w} + (Y^\ell+Z^\ell)\frac{\partial}{\partial v^\ell}\Big\|_{v+w}
			</math>
			and
			:<math>
			\lambda_*\Big(X^k\frac{\partial}{\partial x^k}\Big\|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_v\Big)

			=
			X^k\frac{\partial}{\partial x^k}\Big\|_{\lambda v} + \lambda Y^\ell\frac{\partial}{\partial v^\ell}\Big\|_{\lambda v},
			</math>
			so each fibre (''p''<sub></sub>)<sup>-1</sup>(''X'')⊂''TE'' is a vector space and the triple (''TE'',''p''<sub></sub>,''TM'') is a smooth vector bundle.

			== Linearity of connections on vector bundles ==

			The general [[Ehresmann connection]]
			:<math>
			TE = HE \oplus VE
			</math>
			on a vector bundle (''E'',''p'',''M'') can be characterized in terms of the '''connector map'''
			:<math>
			\kappa:T_vE\to E_{p(v)} \qquad ; \qquad \kappa(X):=\operatorname{vl}_v^{-1}(\operatorname{vpr}X),
			</math>
			where vl<sub>''v''</sub>:''E''→''V''<sub>''v''</sub>''E'' is the [[vector bundle\|vertical lift]], and vpr<sub>''v''</sub>:''T''<sub>''v''</sub>''E''→''V''<sub>''v''</sub>''E'' is the [[Ehresmann connection\|vertical projection]]. The mapping
			:<math>
			\nabla:TM\times\Gamma(E)\to\Gamma(E) \quad ; \quad \nabla_Xv := \kappa(v_*X)
			</math>
			induced by an Ehresmann connection is a [[covariant derivative]] on Γ(''E'') in the sense that
			*<math> \nabla_{X+Y}v = \nabla_X v + \nabla_Y v</math>
			*<math> \nabla_{\lambda X}v=\lambda \nabla_Xv</math>
			*<math> \nabla_X(v+w) = \nabla_X v + \nabla_X w</math>
			*<math> \nabla_X(\lambda v)=\lambda \nabla_Xv</math>
			*<math> \nabla_X(fv) = X[f]v + f\nabla_Xv</math>
			if and only if the connector map is linear with respect to the secondary vector bundle structure (''TE'',''p''<sub>*</sub>,''TM'') on ''TE''. Then the connection is called ''linear''. Note that the connector map is automatically linear with respect to the tangent bundle structure (''TE'',π<sub>''TE''</sub>,''E'').

			== See also ==
			* [[Connection (vector bundle)]]
			* [[Double tangent bundle]]
			* [[Ehresmann connection]]
			* [[Vector bundle]]

			== References ==

			* P.Michor. ''Topics in Differential Geometry,'' American Mathematical Society (2008).

			[[Category:Differential geometry]]
			[[Category:Topology]]
			[[Category:Differential topology]]

en>Cydebot: Robot - Speedily moving category Tiling to :Category:Tessellation per CFDS.

2012-02-16T15:50:30Z

Robot - Speedily moving category Tiling to Category:Tessellation per CFDS.

New page

I'm a 48 years old and work at the university (Biochemistry).<br>In my free time I learn English. I've been twicethere and look forward to go there anytime soon. I love to read, preferably on my kindle. I really love to watch 2 Broke Girls and Doctor Who as well as documentaries about anything astronomical. I like Rock climbing.<br><br>Here is my weblog :: [http://www.ofclan.de/index.php?mod=users&action=view&id=88939 Size guaide mountain bike sizing.]

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en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

Deltahedron: link

en>Cydebot: Robot - Speedily moving category Tiling to :Category:Tessellation per CFDS.