Clos network - Revision history

en>Ray Van De Walker: /* See also */

2014-09-30T23:50:48Z

en>ClueBot NG: Reverting possible vandalism by 103.21.126.76 to version by Monkbot. False positive? Report it. Thanks, ClueBot NG. (1719551) (Bot)

2014-02-26T15:38:40Z

Reverting possible vandalism by 103.21.126.76 to version by Monkbot. False positive? Report it. Thanks, ClueBot NG. (1719551) (Bot)

← Older revision		Revision as of 17:38, 26 February 2014
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	~~{{Unreferenced\|date=November 2006}}~~		Emilia Shryock is my title but you can call me something you like. One of the very best things in the globe for him is to collect badges but he is struggling to discover time for it. For a whilst she's been in South Dakota. Hiring is her working day occupation now and she will not alter it anytime soon.<br><br>my web blog - [http://c-batang.co.kr/?document_srl=485588&mid=levelup_board home std test kit]
	~~In [[Riemannian geometry]], a branch of [[mathematics]], the '''unit tangent bundle''' of a [[Riemannian manifold]] (''M'', ''g''), denoted by UT(''M'') or simply UT''M'',~~ is ~~the unit sphere bundle for the [[tangent bundle]] T(''M'')~~. ~~It is a [[fiber bundle]] over ''M'' whose fiber at each point is~~ the ~~[[unit sphere]]~~ in the ~~tangent bundle:~~

	~~:<math>\mathrm{UT} (M) := \coprod_{x \in M} \left\{ v \in \mathrm{T}_{x} (M) \left\| g_x(v,v) = 1 \right. \right\},</math>~~

	~~where T<sub>''x''</sub>(''M'') denotes the [[tangent space]] to ''M'' at ''x''. Thus, elements of UT(''M'') are pairs (''x'', ''v''), where ''x'' is some point of the manifold and ''v''~~ is ~~some tangent direction (of unit length)~~ to ~~the manifold at ''x''. The unit tangent bundle~~ is ~~equipped with a natural [[projection (mathematics)\|projection]]~~

	~~:<math>\pi : \mathrm{UT} (M) \~~to ~~M,</math>~~
	~~:<math>\pi : (x, v) \mapsto x,</math>~~

	~~which takes each point of the bundle to its base point~~. The fiber ''π''<sup>−1</sup>(''x'') over each point ''x'' ∈ ''M'' is an (''n''−1)-[[hypersphere\|sphere]] '''S'''<sup>''n''−1</sup>, where ''n'' is the dimension of ''M''. The unit tangent bundle is therefore a ~~[[fiber bundle#Sphere bundles\|sphere bundle]] over ''M'' with fiber '''S'''<sup>''n'~~'~~−1</sup>.~~

	~~The definition of unit sphere bundle can easily accommodate [[Finsler manifold]]~~s ~~as well~~. ~~Specifically, if ''M''~~ is a manifold equipped with a Finsler metric ''F'' : T''M'' → '''R''', then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at ''x'' is the indicatrix of ''F'':
	~~:<math>\mathrm{UT}_x (M) = \left\{ v \in \mathrm{T}_{x} (M) \left\| F(v) = 1 \right. \right\}~~.<~~/math~~>

	If ''M'' is an infinite-dimensional manifold (for example, a [[Banach manifold\|Banach]], [[Fréchet manifold\|Fréchet]] or [[Hilbert manifold]]), then UT(''M'') can still be thought of as the unit sphere bundle for the tangent bundle T(''M''), but the fiber ''π''<sup>−1<~~/sup~~>~~(''x'') over ''x'' is then the infinite~~-~~dimensional unit sphere in the tangent space.~~

	~~==Structures==~~
	~~The unit tangent bundle carries a variety of differential geometric structures. The metric on ''M'' induces a [~~[~~contact structure]] on UT''M''. This is given in terms of a tautological one-form θ, defined at a point ''u'' of UT''M'' (a unit tangent vector of ''M'') by~~
	:~~<math>\theta_u(v) = g(u,\pi_* v)\,</math>~~
	~~where π<sub>*<~~/~~sub> is the [[pushforward (differential)\|pushforward]] along π of the vector ''v'' ∈ T<sub>''u''<~~/~~sub>UT''M''.~~

	~~Geometrically, this contact structure can be regarded as the distribution of (2''n''−2)~~-planes which, at the unit vector ''u'', is the pullback of the orthogonal complement of ''u'' in the tangent space of ''M''. This is a contact structure, for the fiber of UT''M'' is obviously an integral manifold (the vertical bundle is everywhere in the kernel of θ), and the remaining tangent directions are filled out by moving up the fiber of UT''M''. Thus the maximal integral manifold of θ is (an open set of) ''M'' itself.

	~~On a Finsler manifold, the contact form is defined by the analogous formula~~
	~~:<math>\theta_u(v) = g_u(u,\pi_*v)\,</math>~~
	~~where ''g''<sub>''u''</sub> is the fundamental tensor (the [[Hessian matrix\|hessian]] of the Finsler metric)~~. Geometrically, the associated distribution of hyperplanes at the point ''u'' ∈ UT<sub>''x''</sub>''M'' is the inverse image under π<sub>*</sub> of the tangent hyperplane to the unit sphere in T<sub>''x''</sub>''M'' at ''u''.

	~~The [[volume form]] θ∧''d''θ<sup>''n''−1<~~/sup> defines a [[measure (mathematics)\|measure]] on ''M'', known as the '''kinematic measure''', or '''Liouville measure''', that is invariant under the [[geodesic#Geodesic flow\|geodesic flow]] of ''M''. As a [[Radon measure]], the kinematic measure μ is defined on compactly supported continuous functions ''ƒ'' on UT''M'' by
	~~:<math>\int_{UTM} f\,d\mu = \int_M dV(p) \int_{UT_pM} \left.f\right\|_{UT_pM}\,d\mu_p</math>~~
	~~where d''V'' is the [[volume element]] on ''M'', and μ<sub>''p''</sub> is the standard rotationally-invariant [[Borel measure]] on the Euclidean sphere UT<sub>''p''</sub>''M''.~~

	~~The [[Levi-Civita connection]] of ''M'' gives rise to a splitting of the tangent bundle~~
	~~:<math>T(UTM)~~ = ~~H\oplus V</math>~~
	~~into a vertical space ''V''~~&~~nbsp;~~= kerπ<sub></sub> and horizontal space ''H'' on which π<sub></sub> is a [[linear isomorphism]] at each point of UT''M''. This splitting induces a metric on UT''M'' by declaring that this splitting be an orthogonal direct sum, and defining the metric on ''H'' by the pullback:
	~~:<math>g_H(v,w) = g(v,w),\quad v,w\in H</math>~~
	and defining the metric on ''V'' as the induced metric from the embedding of the fiber UT<sub>''x''</sub>''M'' into the [[Euclidean space]] T<sub>''x''</sub>''M''. Equipped with this metric and contact form, UT''M'' becomes a [[Sasakian manifold]].

	~~{{DEFAULTSORT:Unit Tangent Bundle}}~~
	~~[[Category:Differential topology]]~~
	~~[[Category:Ergodic theory]]~~
	~~[[Category:Fiber bundles]]~~
	~~[[Category:Riemannian geometry]~~]

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-02-03T22:30:59Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 00:30, 4 February 2014
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	~~Greetings~~. ~~The author~~'~~s title~~ is ~~Phebe and she feels comfy when people use~~ the ~~complete name~~. ~~Hiring has been my profession for some time but I~~'~~ve already utilized for another 1~~. ~~My family members lives in Minnesota~~ and ~~my family members loves it~~. ~~What I adore performing~~ is ~~playing baseball but I haven't produced~~ a ~~dime at home~~ [~~http~~://~~women~~.~~Webmd~~.~~com~~/~~features~~/~~could~~-~~you-have-std-not-know std test~~] ~~with it.~~<br><br>~~Feel~~ ~~home~~ [~~http~~://~~www~~.1a-~~pornotube~~.~~com~~/~~blog~~/~~84958 std testing~~ at ~~home~~] ~~test kit free to visit my web page~~; [~~http~~://~~www.hamcass.org~~/~~index~~.~~php?document_srl~~=~~268708~~&~~mid~~=~~gido~~ at ~~home std testing]~~ ([~~http:~~/~~/www~~.~~adosphere~~.~~com/poyocum simply click the up coming internet site~~])		{{Unreferenced\|date=November 2006}}
			In [[Riemannian geometry]], a branch of [[mathematics]], the '''unit tangent bundle''' of a [[Riemannian manifold]] (''M'', ''g''), denoted by UT(''M'') or simply UT''M'', is the unit sphere bundle for the [[tangent bundle]] T(''M''). It is a [[fiber bundle]] over ''M'' whose fiber at each point is the [[unit sphere]] in the tangent bundle:

			:<math>\mathrm{UT} (M) := \coprod_{x \in M} \left\{ v \in \mathrm{T}_{x} (M) \left\| g_x(v,v) = 1 \right. \right\},</math>

			where T<sub>''x''</sub>(''M'') denotes the [[tangent space]] to ''M'' at ''x''. Thus, elements of UT(''M'') are pairs (''x'', ''v''), where ''x'' is some point of the manifold and ''v'' is some tangent direction (of unit length) to the manifold at ''x''. The unit tangent bundle is equipped with a natural [[projection (mathematics)\|projection]]

			:<math>\pi : \mathrm{UT} (M) \to M,</math>
			:<math>\pi : (x, v) \mapsto x,</math>

			which takes each point of the bundle to its base point. The fiber ''π''<sup>−1</sup>(''x'') over each point ''x'' ∈ ''M'' is an (''n''−1)-[[hypersphere\|sphere]] '''S'''<sup>''n''−1</sup>, where ''n'' is the dimension of ''M''. The unit tangent bundle is therefore a [[fiber bundle#Sphere bundles\|sphere bundle]] over ''M'' with fiber '''S'''<sup>''n''−1</sup>.

			The definition of unit sphere bundle can easily accommodate [[Finsler manifold]]s as well. Specifically, if ''M'' is a manifold equipped with a Finsler metric ''F'' : T''M'' → '''R''', then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at ''x'' is the indicatrix of ''F'':
			:<math>\mathrm{UT}_x (M) = \left\{ v \in \mathrm{T}_{x} (M) \left\| F(v) = 1 \right. \right\}.</math>

			If ''M'' is an infinite-dimensional manifold (for example, a [[Banach manifold\|Banach]], [[Fréchet manifold\|Fréchet]] or [[Hilbert manifold]]), then UT(''M'') can still be thought of as the unit sphere bundle for the tangent bundle T(''M''), but the fiber ''π''<sup>−1</sup>(''x'') over ''x'' is then the infinite-dimensional unit sphere in the tangent space.

			==Structures==
			The unit tangent bundle carries a variety of differential geometric structures. The metric on ''M'' induces a [[contact structure]] on UT''M''. This is given in terms of a tautological one-form θ, defined at a point ''u'' of UT''M'' (a unit tangent vector of ''M'') by
			:<math>\theta_u(v) = g(u,\pi_* v)\,</math>
			where π<sub>*</sub> is the [[pushforward (differential)\|pushforward]] along π of the vector ''v'' ∈ T<sub>''u''</sub>UT''M''.

			Geometrically, this contact structure can be regarded as the distribution of (2''n''−2)-planes which, at the unit vector ''u'', is the pullback of the orthogonal complement of ''u'' in the tangent space of ''M''. This is a contact structure, for the fiber of UT''M'' is obviously an integral manifold (the vertical bundle is everywhere in the kernel of θ), and the remaining tangent directions are filled out by moving up the fiber of UT''M''. Thus the maximal integral manifold of θ is (an open set of) ''M'' itself.

			On a Finsler manifold, the contact form is defined by the analogous formula
			:<math>\theta_u(v) = g_u(u,\pi_*v)\,</math>
			where ''g''<sub>''u''</sub> is the fundamental tensor (the [[Hessian matrix\|hessian]] of the Finsler metric). Geometrically, the associated distribution of hyperplanes at the point ''u'' ∈ UT<sub>''x''</sub>''M'' is the inverse image under π<sub>*</sub> of the tangent hyperplane to the unit sphere in T<sub>''x''</sub>''M'' at ''u''.

			The [[volume form]] θ∧''d''θ<sup>''n''−1</sup> defines a [[measure (mathematics)\|measure]] on ''M'', known as the '''kinematic measure''', or '''Liouville measure''', that is invariant under the [[geodesic#Geodesic flow\|geodesic flow]] of ''M''. As a [[Radon measure]], the kinematic measure μ is defined on compactly supported continuous functions ''ƒ'' on UT''M'' by
			:<math>\int_{UTM} f\,d\mu = \int_M dV(p) \int_{UT_pM} \left.f\right\|_{UT_pM}\,d\mu_p</math>
			where d''V'' is the [[volume element]] on ''M'', and μ<sub>''p''</sub> is the standard rotationally-invariant [[Borel measure]] on the Euclidean sphere UT<sub>''p''</sub>''M''.

			The [[Levi-Civita connection]] of ''M'' gives rise to a splitting of the tangent bundle
			:<math>T(UTM) = H\oplus V</math>
			into a vertical space ''V'' = kerπ<sub></sub> and horizontal space ''H'' on which π<sub></sub> is a [[linear isomorphism]] at each point of UT''M''. This splitting induces a metric on UT''M'' by declaring that this splitting be an orthogonal direct sum, and defining the metric on ''H'' by the pullback:
			:<math>g_H(v,w) = g(v,w),\quad v,w\in H</math>
			and defining the metric on ''V'' as the induced metric from the embedding of the fiber UT<sub>''x''</sub>''M'' into the [[Euclidean space]] T<sub>''x''</sub>''M''. Equipped with this metric and contact form, UT''M'' becomes a [[Sasakian manifold]].

			{{DEFAULTSORT:Unit Tangent Bundle}}
			[[Category:Differential topology]]
			[[Category:Ergodic theory]]
			[[Category:Fiber bundles]]
			[[Category:Riemannian geometry]]

en>A5b: link to ru:~

2012-08-12T23:39:50Z

link to ru:~

New page

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← Older revision		Revision as of 01:50, 1 October 2014
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	~~Emilia Shryock~~ is my ~~title but you can call me something you like~~. ~~One of the very best things in the globe~~ for ~~him~~ is to ~~collect badges~~ but ~~he is struggling to discover time for~~ it. ~~For a whilst she's been in South Dakota. Hiring~~ is ~~her working day occupation now and she will not alter it anytime soon~~.<br><br>my ~~web~~ blog - [http://c-~~batang.co.kr~~/~~?document_srl=485588&mid=levelup_board home std test kit~~]		Nice to satisfy you, my name is Figures Held although I don't really like becoming called like that. [http://homestdtestkit.org/instant-std-test-kits/ South Dakota] is exactly where me and my spouse live and my family members loves it. The preferred pastime for my kids and me is to perform baseball [http://www.1a-pornotube.com/user/GCalvert at home std testing] home std test but I haven't produced a dime [http://www.alemcheap.fi/people/anhipkiss at home std testing] with it. Bookkeeping is what I do.<br><br>Feel free to surf to my blog :: at home std test; [http://carnavalsite.com/demo-page-1/solid-advice-in-relation-to-yeast-infection/ go to website],