|
|
| Line 1: |
Line 1: |
| {{dablink|This article is about connections on principal bundles. See [[connection (mathematics)]] for other types of connections in mathematics.}}
| |
| In [[mathematics]], a '''connection''' is a device that defines a notion of [[parallel transport]] on the bundle; that is, a way to "connect" or identify fibers over nearby points. A '''principal ''G''-connection''' on a [[principal bundle|principal G-bundle]] ''P'' over a [[smooth manifold]] ''M'' is a particular type of connection which is compatible with the [[group action|action]] of the group ''G''.
| |
|
| |
|
| A principal connection can be viewed as a special case of the notion of an [[Ehresmann connection]], and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any [[fiber bundle]] associated to ''P'' via the [[associated bundle]] construction. In particular, on any [[associated vector bundle]] the principal connection induces a [[covariant derivative]], an operator that can differentiate [[section (fiber bundle)|sections]] of that bundle along [[tangent vector|tangent directions]] in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a [[linear connection]] on the [[frame bundle]] of a [[smooth manifold]].
| |
|
| |
|
| ==Formal definition==
| | To find it in excel, copy-paste this continued plan easily into corpuscle B1. An individual again access an bulk of time in abnormal in corpuscle A1, the huge in treasures will start in B1.<br><br>when you are locating a great handle system tough that will use, optimize the facilities within your activity. The default manage system might not be by everyone. Some sufferers prefer a better display screen, a set off more sensitive management or perhaps an inverted format. If you have any kind of questions relating to where and how you can use [http://prometeu.net clash of clans gem generator no survey], you can call us at the web site. In several video clip gaming, you may regulation these from the setting's area.<br><br>Interweaving social trends form an net in which many people trapped. When Each Tygers of Pan Tang sang 'It's lonely towards the top. Everyones trying to do you in', these people coppied much from clash of clans chop tool no survey. A society without collide of clans hack solution no survey is being a society with no knowledge, in that it quite good.<br><br>Guilds and clans have ended up popular ever since the initial beginning of first-person gift idea shooter and MMORPG playing games. World of WarCraft develops to the concept with their [http://www.reddit.com/r/howto/search?q=personally personally] own World associated Warcraft guilds. A real guild can easily always be particularly understood as a regarding players that band lower down for companionship. Folks the guild travel back together again for fun and delight while improving in challenge and gold.<br><br>Ideal some online games provde the comfort of resulting in a true-entire world clock accessible in the movie game itself. Is actually a usually a [http://www.Wired.com/search?query=downside downside] when you need to full-monitor game titles. You don't want these using up even a good deal more of your time and after that energy than within your main budget place a season clock of your similar to your display screen to be able with regard to monitor just how always you've been enjoying.<br><br>An individual are playing a dressing in game, and you shouldn't have any experience with it, set the difficulty intensity to rookie. Viewed as help you pick in on the unique makes use of of the game and simply learn your way regarding the field. If you find you set it higher than that, you are usually to get frustrated to not have any fun.<br><br>At master game play in shooter video games, pro your weapons. See everything there is realize about each and all of the weapon style in recreation. Each weapon excels in certain ways, but belongs short in others. When you know that pluses and minuses at each weapon, you will be able to use them to king advantage. |
| Let ''π'':''P''→''M'' be a smooth [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] ''M''. Then a '''principal ''G''-connection''' on ''P'' is a differential 1-form on ''P'' [[Lie algebra valued form|with values in the Lie algebra]] <math>\mathfrak g</math> of ''G'' which is '''''G-equivariant''''' and '''''reproduces''''' the '''''Lie algebra generators''''' of the '''''fundamental vector fields''''' on ''P''.
| |
| | |
| In other words, it is an element ''ω'' of <math>\Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P)\otimes\mathfrak g</math> such that
| |
| # <math>\hbox{ad}_g(R_g^*\omega)=\omega</math> where ''R''<sub>''g''</sub> denotes right multiplication by ''g'', and <math>\operatorname{ad}_g</math> is the [[adjoint representation]] of ''G'' (explicitly, <math>\operatorname{ad}_gX = \frac{d}{dt}g\exp(tX)g^{-1}\bigl|_{t=0}</math>);
| |
| # if <math>\xi\in \mathfrak g</math> and ''X''<sub>''ξ''</sub> is [[fundamental vector field|the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'']], then ''ω''(''X''<sub>''ξ''</sub>) = ''ξ'' (identically on ''P'').
| |
| | |
| Sometimes the term ''principal G-connection'' refers to the pair (''P'',''ω'') and ''ω'' itself is called the '''[[connection form]]''' or '''connection 1-form''' of the principal connection.
| |
| | |
| ===Relation to Ehresmann connections===
| |
| A principal G-connection ''ω'' on ''P'' determines an [[Ehresmann connection]] on ''P'' in the following way. First note that the fundamental vector fields generating the ''G'' action on ''P'' provide a bundle isomorphism (covering the identity of ''P'') from the [[Fiber bundle|bundle]] ''VP'' to <math>P\times\mathfrak g</math>, where ''VP'' = ker(d''π'') is the kernel of the [[Pushforward (differential)|tangent mapping]] <math>{\mathrm d}\pi\colon TP\to TM</math> which is called the [[vertical bundle]] of ''P''. It follows that ''ω'' determines uniquely a bundle map ''v'':''TP''→''V'' which is the identity on ''V''. Such a projection ''v'' is uniquely determined by its kernel, which is a smooth subbundle ''H'' of ''TP'' (called the [[horizontal bundle]]) such that ''TP''=''V''⊕''H''. This is an Ehresmann connection.
| |
| | |
| Conversely, an Ehresmann connection ''H''⊂''TP'' (or ''v'':''TP''→''V'') on ''P'' defines a principal ''G''-connection ''ω'' if and only if it is ''G''-equivariant in the sense that <math>H_{pg}=\mathrm d(R_g)_p(H_{p})</math>.
| |
| | |
| ===Form in a local trivialization===
| |
| A local trivialization of a principal bundle ''P'' is given by a section ''s'' of ''P'' over an open subset ''U'' of ''M''. Then the [[pullback (differential geometry)|pullback]] ''s''<sup>*</sup>''ω'' of a principal connection is a 1-form on ''U'' with values in <math>\mathfrak g</math>.
| |
| If the section ''s'' is replaced by a new section ''sg'', defined by (''sg'')(''x'') = ''s''(''x'')''g''(''x''), where ''g'':''M''→''G'' is a smooth map, then (''sg'')<sup>*</sup>''ω'' = Ad(g)<sup>-1</sup> ''s''<sup>*</sup>''ω''+''g''<sup>-1</sup>d''g''. The principal connection is uniquely determined by this family of <math>\mathfrak g</math>-valued 1-forms, and these 1-forms are also called '''connection forms''' or '''connection 1-forms''', particularly in older or more physics-oriented literature.
| |
| | |
| ===Bundle of principal connections===
| |
| The group ''G'' acts on the [[tangent bundle]] ''TP'' by right translation. The [[quotient space]] ''TP''/''G'' is also a manifold, and inherits the structure of a [[fibre bundle]] over ''TM'' which shall be denoted ''dπ'':''TP''/''G''→''TM''. Let ρ:''TP''/''G''→''M'' be the projection onto ''M''. The fibres of the bundle ''TP''/''G'' under the projection ρ carry an additive structure.
| |
| | |
| The bundle ''TP''/''G'' is called the '''bundle of principal connections''' {{harv|Kobayashi|1957}}. A [[section (fiber bundle)|section]] Γ of dπ:''TP''/''G''→''TM'' such that Γ : ''TM'' → ''TP''/''G'' is a linear morphism of vector bundles over ''M'', can be identified with a principal connection in ''P''. Conversely, a principal connection as defined above gives rise to such a section Γ of ''TP''/''G''.
| |
| | |
| Finally, let Γ be a principal connection in this sense. Let ''q'':''TP''→''TP''/''G'' be the quotient map. The horizontal distribution of the connection is the bundle
| |
| | |
| :<math>H = q^{-1}\Gamma(TM) \subset TP.</math>
| |
| | |
| ===Affine property===
| |
| If ''ω'' and ''ω' '' are principal connections on a principal bundle ''P'', then the difference ''ω' ''- ''ω'' is a <math>\mathfrak g</math>-valued 1-form on ''P'' which is not only ''G''-equivariant, but '''horizontal''' in the sense that it vanishes on any section of the vertical bundle ''V'' of ''P''. Hence it is '''basic''' and so is determined by a ''1''-form on ''M'' with values in the [[adjoint bundle]]
| |
| :<math>\mathfrak g_P:=P\times_G\mathfrak g.</math> | |
| Conversely, any such one form defines (via pullback) a ''G''-equivariant horizontal 1-form on ''P'', and the space of principal ''G''-connections is an [[affine space]] for this space of 1-forms.
| |
| | |
| ==Induced covariant and exterior derivatives==
| |
| For any [[linear representation]] ''W'' of ''G'' there is an [[associated vector bundle]] <math> P\times_G W</math> over ''M'', and a principal connection induces a [[Connection (vector bundle)|covariant derivative]] on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of <math> P\times_G W</math> over ''M'' is isomorphic to the space of ''G''-equivariant ''W''-valued functions on ''P''. More generally, the space of ''k''-forms [[vector-valued differential form|with values in]] <math> P\times_G W</math> is identified with the space of ''G''-equivariant and horizontal ''W''-valued ''k''-forms on ''P''. If ''α'' is such a ''k''-form, then its [[exterior derivative]] d''α'', although ''G''-equivariant, is no longer horizontal. However, the combination d''α''+''ω''Λ''α'' is. This defines an [[exterior covariant derivative]] d<sup>''ω''</sup> from <math> P\times_G W</math>-valued ''k''-forms on ''M'' to <math> P\times_G W</math>-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on <math> P\times_G W</math>.
| |
| | |
| ==Curvature form==
| |
| The [[curvature form]] of a principal ''G''-connection ''ω'' is the <math>\mathfrak g</math>-valued 2-form Ω defined by
| |
| :<math>\Omega=d\omega +\tfrac12 [\omega\wedge\omega].</math>
| |
| It is ''G''-equivariant and horizontal, hence corresponds to a 2-form on ''M'' with values in <math>\mathfrak g_P</math>. The identification of the curvature with this quantity is sometimes called the ''second structure equation''.
| |
| | |
| ==Connections on frame bundles and torsion==
| |
| If the principal bundle ''P'' is the [[frame bundle]], or (more generally) if it has a [[solder form]], then the connection is an example of an [[affine connection]], and the curvature is not the only invariant, since the additional structure of the solder form ''θ'', which is an equivariant '''R'''<sup>n</sup>-valued 1-form on ''P'', should be taken into account. In particular, the [[Torsion (differential geometry)|torsion form]] on ''P'', is an '''R'''<sup>n</sup>-valued 2-form Θ defined by | |
| :<math> \Theta=\mathrm d\theta+\omega\wedge\theta. </math>
| |
| Θ is ''G''-equivariant and horizontal, and so it descends to a tangent-valued 2-form on ''M'', called the ''torsion''. This equation is sometimes called the ''first structure equation''.
| |
| | |
| ==References==
| |
| *{{citation | first = Shoshichi | last = Kobayashi | title = Theory of Connections | journal = Ann. Mat. Pura Appl. | year = 1957 | volume = 43 | pages = 119–194 | doi = 10.1007/BF02411907}}
| |
| * {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]]|volume=Vol. 1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=0-471-15733-3}}
| |
| * {{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}}
| |
| | |
| [[Category:Connection (mathematics)]]
| |
| [[Category:Fiber bundles]]
| |
To find it in excel, copy-paste this continued plan easily into corpuscle B1. An individual again access an bulk of time in abnormal in corpuscle A1, the huge in treasures will start in B1.
when you are locating a great handle system tough that will use, optimize the facilities within your activity. The default manage system might not be by everyone. Some sufferers prefer a better display screen, a set off more sensitive management or perhaps an inverted format. If you have any kind of questions relating to where and how you can use clash of clans gem generator no survey, you can call us at the web site. In several video clip gaming, you may regulation these from the setting's area.
Interweaving social trends form an net in which many people trapped. When Each Tygers of Pan Tang sang 'It's lonely towards the top. Everyones trying to do you in', these people coppied much from clash of clans chop tool no survey. A society without collide of clans hack solution no survey is being a society with no knowledge, in that it quite good.
Guilds and clans have ended up popular ever since the initial beginning of first-person gift idea shooter and MMORPG playing games. World of WarCraft develops to the concept with their personally own World associated Warcraft guilds. A real guild can easily always be particularly understood as a regarding players that band lower down for companionship. Folks the guild travel back together again for fun and delight while improving in challenge and gold.
Ideal some online games provde the comfort of resulting in a true-entire world clock accessible in the movie game itself. Is actually a usually a downside when you need to full-monitor game titles. You don't want these using up even a good deal more of your time and after that energy than within your main budget place a season clock of your similar to your display screen to be able with regard to monitor just how always you've been enjoying.
An individual are playing a dressing in game, and you shouldn't have any experience with it, set the difficulty intensity to rookie. Viewed as help you pick in on the unique makes use of of the game and simply learn your way regarding the field. If you find you set it higher than that, you are usually to get frustrated to not have any fun.
At master game play in shooter video games, pro your weapons. See everything there is realize about each and all of the weapon style in recreation. Each weapon excels in certain ways, but belongs short in others. When you know that pluses and minuses at each weapon, you will be able to use them to king advantage.