en>Melcombe: grammar, and add {{expert-subject}}

2013-04-20T15:05:17Z

grammar, and add {{expert-subject}}

← Older revision		Revision as of 17:05, 20 April 2013
Line 1:		Line 1:
	~~Alyson~~ is the title ~~people use~~ to ~~contact me and I think it seems quite good when you say it~~. ~~My spouse~~ and ~~I live~~ in ~~Mississippi but now I~~'~~m psychic solutions by lynne~~ ([~~http~~://~~chorokdeul~~.co.kr/~~index~~.~~php?document_srl~~=~~324263&mid~~=~~customer21 Enter your text here and click~~ the ~~"Remove Empty Lines" button above~~.]) ~~contemplating other choices~~. ~~What me~~ and ~~my family members adore is~~ to ~~climb~~ but ~~I'm considering on starting some thing new~~. ~~Distributing manufacturing~~ is ~~exactly where my main earnings comes~~ from ~~and~~ it'~~s something I really enjoy~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~online psychic reading (~~[~~http:~~//~~cpacs~~.~~org~~/~~index~~.~~php?document_srl~~=~~90091&mid~~=~~board_zTGg26 cpacs~~.~~org~~]) ~~my website online psychic chat - [https~~://www~~-ocl.gist~~.ac.kr/~~work~~/xe/~~?document_srl~~=~~605236 www~~-~~ocl~~.~~gist~~.ac.kr] -		In [[differential geometry]], in the category of [[differentiable manifold]]s, a '''fibered manifold''' is a [[surjective]] [[submersion (mathematics)\|submersion]] <ref>{{citation\|last1 = Kolář\|first1=Ivan\|last2=Michor\|first2=Peter\|last3=Slovák\|first3=Jan\|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf\|format=PDF\|title=Natural operators in differential geometry\|year = 1993\|publisher = Springer-Verlag}}. (page 11)
			</ref> <math>\pi \colon E \to B\, ,</math> i.e. a surjective differentiable mapping <math>\pi \colon E \to B\,</math> such that at each point <math>y\in E</math> the tangent mapping <math>T_y\pi \colon T_{y}E \to T_{\pi(y)}B</math> is surjective (equivalently its rank equals dim ''B'').

			==Formal definition==
			A triple <math>(E,\pi,B)\, ,</math> where ''E'' and ''B'' are differentiable manifolds and <math>\pi \colon E \to B\,</math> is a surjective submersion, is called a '''fibered manifold'''.<ref>{{citation\|last1 = Krupka\|first1=Demeter\|last2=Janyška\|first2=Josef\|title=Lectures on differential invariants\|year = 1990\|publisher = Univerzita J. E. Purkyně V Brně\|isbn=80-210-0165-8}}. (page 47)
			</ref> ''E'' is called the '''total space''', ''B'' is called the '''base'''.

			==Examples==
			* Every differentiable [[fiber bundle]] is a '''fibered manifold'''.
			* Every differentiable [[covering space]] is a '''fibered manifold''' with discrete fiber.
			* In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle <math>({\mathbb {S}^1}\times {\mathbb {R}^1},{\mathrm {pr}_1},{\mathbb {S}^1})\, ,</math> and deleting two points in to two different fibers over the base manifold <math>{\mathbb {S}^1}\,</math>.The result is a new fibered manifold where all the fibers except two are connected.

			==Properties==
			Let <math>B</math> (resp. <math>E</math>) be an ''n''-dimensional (resp. ''p''-dimensional) manifold. A '''fibered manifold''' <math>(E,\pi,B)\,</math> admits '''fiber charts'''. We say that a [[chart]] <math>(V,\psi)\,</math> on <math>E\,</math> is a '''fiber chart''', or is '''adapted''' to the surjective submersion <math>\pi \colon E \to B\, ,</math> if there exists a chart <math>(U,\varphi)\,</math> on <math>B\,</math> such that <math>U=\pi (V)\,</math> and
			: <math>u^1=x^1\circ \pi,\,u^2=x^2\circ \pi,\,\dots,\,u^n=x^n\circ \pi\, ,</math>
			where <math>\psi=(u^1,\dots,u^n,y^1,\dots,y^{p-n})\,</math> with <math>y_{0}\in V\,</math> and <math>\varphi=(x^1,\dots,x^n)\,</math> with <math>\pi(y_{0})\in U\, .</math> The above fiber chart condition may be equivalently expressed by
			: <math>\varphi\circ\pi={\mathrm {pr}_1}\circ\psi\, ,</math>
			where <math>{\mathrm {pr}_1} \colon {\mathbb R^n}\times{\mathbb R^{p-n}} \to {\mathbb R^n}\,</math> is the first projection. The chart <math>(U,\varphi)\,</math> is then obviously unique. In view of the above property, the coordinates of a fiber chart <math>(V,\psi)\,</math> are usually denoted by <math>\psi = (x^i,y^{\sigma})\, ,</math> where <math>i\in \{1,\dots,n\}\, ,</math> <math>\sigma\in \{1,\dots,m\}\, ,</math> <math>m=p-n\, ;</math> the coordinates of the corresponding [[chart]] <math>(U,\varphi)\,</math> on <math>B\,</math> are then denoted, with the obvious convention, by <math>\varphi = (x^i)\, ,</math> where <math>i\in \{1,\dots,n\}\, .</math>

			Any surjective submersion <math>\pi \colon E \to B\,</math> is open: for each open <math>V\subset E\,</math> the set <math>\pi(V)\subset B\,</math> is open in <math>B\,</math>.

			A fibered manifold admits local sections: For each <math>y\in E\,</math> there is an open neighborhood <math>U\,</math> of <math>\pi(y)\,</math> in <math>B\,</math> and a smooth mapping <math>s\colon U\to E\,</math> with <math>\pi\circ s={\mathrm {Id}_U}\,</math> and <math>s(\pi(y))=y\,</math>.

			==History==
			In [[topology]], the words '''fiber''' ('''Faser''' in German) and '''fiber space''' ('''gefaserter Raum''') appeared for the first time in a paper by [[Herbert Seifert\|Seifert]] in 1932,<ref>{{cite journal\|title=Topologie dreidimensionaler geschlossener Räume\|author=H. Seifert\|journal=Acta Math.\|volume=60\|year=1932\|pages=147–238}}</ref> but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the '''base space''' (topological space) of a fiber (topological) space ''E'' was not part of the structure, but derived from it as a quotient space of ''E''. The first definition of '''fiber space''' is given by [[Hassler Whitney]] in 1935 <ref>{{cite journal\|title=Sphere spaces\|author=H. Whitney\|journal=Proc. Nat. Acad. Sci. USA\|volume=21\|year=1935\|pages=464–468}}</ref> under the name '''sphere space''', but in 1940 Whitney changed the name to '''sphere bundle'''.<ref>{{cite journal\|title=On the theory of sphere bundles\|author=H. Whitney\|journal=Proc. Nat. Acad. Sci. USA\|volume=26\|year=1940\|pages=148–153}}</ref>

			The theory of fibered spaces, of which [[vector bundle]]s, [[principal bundle]]s, topological [[fibration]]s and fibered manifolds are a special case, is attributed to [[Herbert Seifert\|Seifert]], [[Heinz Hopf\|Hopf]], [[Jacques Feldbau\|Feldbau]],<ref>{{cite journal\|title=Sur la classification des espaces fibrés\|author=J. Feldbau\|journal=C. R. Acad. Sci. Paris\|volume=208\|year=1939\|pages=1621–1623}}</ref> [[Hassler Whitney\|Whitney]], [[Norman Steenrod\|Steenrod]], [[Charles Ehresmann\|Ehresmann]],<ref>{{cite journal\|title=Sur la théorie des espaces fibrés\|author=C. Ehresmann\|journal=Coll. Top. alg. Paris\|volume=C.N.R.S.\|year=1947\|pages=3–15}}</ref><ref>{{cite journal\|title=Sur les espaces fibrés différentiables\|author=C. Ehresmann\|journal=C. R. Acad. Sci. Paris\|volume=224\|year=1947\|pages=1611–1612}}</ref><ref>{{cite journal\|title=Les prolongements d'un espace fibré différentiable\|author=C. Ehresmann\|journal=C. R. Acad. Sci. Paris\|volume=240\|year=1955\|pages=1755–1757}}</ref> [[Jean-Pierre Serre\|Serre]],<ref>{{cite journal\|title=Homologie singulière des espaces fibrés. Applications\|author=J.-P. Serre\|journal=Ann. of Math.\|volume=54\|year=1951\|pages=425–505}}</ref> and others.

			==See also==
			* [[Covering space]]
			* [[Fiber bundle]]
			* [[Fibration]]
			* [[Seifert fiber space]]
			* [[Connection (fibred manifold)]]

			==Notes==
			{{Reflist}}

			==References==
			* {{citation\|last1 = Kolář\|first1=Ivan\|last2=Michor\|first2=Peter\|last3=Slovák\|first3=Jan\|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf\|format=PDF\|title=Natural operators in differential geometry\|year = 1993\|publisher = Springer-Verlag}}
			* {{citation\|last1 = Krupka\|first1=Demeter\|last2=Janyška\|first2=Josef\|title=Lectures on differential invariants\|year = 1990\|publisher = Univerzita J. E. Purkyně V Brně\|isbn=80-210-0165-8}}
			* {{citation\|last1 = Saunders\|first1=D.J.\|title=The geometry of jet bundles\|year = 1989\|publisher = Cambridge University Press\|isbn=0-521-36948-7}}

			[[Category:Manifolds]]
			[[Category:Fiber bundles]]

en>Naught101: link up

2012-04-09T13:49:30Z

link up

New page

Alyson is the title people use to contact me and I think it seems quite good when you say it. My spouse and I live in Mississippi but now I'm psychic solutions by lynne ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 Enter your text here and click the "Remove Empty Lines" button above.]) contemplating other choices. What me and my family members adore is to climb but I'm considering on starting some thing new. Distributing manufacturing is exactly where my main earnings comes from and it's something I really enjoy.<br><br>Feel free to surf to online psychic reading ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 cpacs.org]) my website online psychic chat - [https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 www-ocl.gist.ac.kr] -

Mean squared prediction error - Revision history

en>Melcombe: grammar, and add {{expert-subject}}

en>Naught101: link up