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In [[mathematics]], a '''principal bundle'''<ref>{{cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | publisher = Princeton University Press | location = Princeton | year = 1951 | isbn = 0-691-00548-6}} page 35
</ref><ref>{{cite book | last = Husemoller | first = Dale | title = Fibre Bundles | publisher = Springer | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}} page 42
</ref><ref>{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year = 1997 | isbn = 0-387-94732-9}} page 37
</ref><ref>{{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise |author2-link=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}} page 370</ref> is a mathematical object which formalizes some of the essential features of the [[Cartesian product]] ''X'' &times; ''G'' of a space ''X'' with a [[group (mathematics)|group]] ''G''. In the same way as with the Cartesian product, a principal bundle ''P'' is equipped with
# An [[group action|action]] of ''G'' on ''P'', analogous to (''x'',''g'')''h'' = (''x'', ''gh'') for a product space.
# A projection onto ''X''. For a product space, this is just the projection onto the first factor, (''x'',''g'') → ''x''.
Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (''x'',''e'').  Likewise, there is not generally a projection onto ''G'' generalizing the projection onto the second factor, ''X'' &times; ''G'' → ''G'' which exists for the Cartesian product. They may also have a complicated [[topology]], which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.
 
A common example of a principal bundle is the [[frame bundle]] F''E'' of a [[vector bundle]] ''E'', which consists of all ordered [[basis of a vector space|bases]] of the vector space attached to each point. The group ''G'' in this case is the [[general linear group]], which acts in the usual{{clarify|date=April 2013}} way on ordered bases.  Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.
 
Principal bundles have important applications in [[topology]] and [[differential geometry]]. They have also found application in [[physics]] where they form part of the foundational framework of [[gauge theory|gauge theories]]. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group ''G'' determine a unique principal ''G''-bundle from which the original bundle can be reconstructed.
 
==Formal definition==
 
A principal ''G''-bundle, where ''G'' denotes any [[topological group]], is a [[fiber bundle]] ''π'' : ''P'' → ''X'' together with a [[continuous (topology)|continuous]] [[group action|right action]] ''P'' × ''G'' → ''P'' such that ''G'' preserves the fibers of ''P'' (i.e. if ''y'' ∈ P<sub>''x''</sub> then ''yg'' ∈ P<sub>''x''</sub> for all ''g'' ∈ ''G'') and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group ''G'' itself. Frequently, one requires the base space ''X'' to be [[Hausdorff space|Hausdorff]] and possibly [[paracompact]].
 
Since the group action preserves the fibers of ''π'' : ''P'' → ''X'' and acts transitively, it follows that the [[orbit (group theory)|orbits]] of the ''G''-action are precisely these fibers and the orbit space ''P''/''G'' is [[homeomorphic]] to the base space ''X''. Because the action is free, the fibers have the structure of [[torsor|''G''-torsors]]. A ''G''-torsor is a space which is homeomorphic to ''G'' but lacks a group structure since there is no preferred choice of an [[identity element]].
 
An equivalent definition of a principal ''G''-bundle is as a ''G''-bundle ''π'' : ''P'' → ''X'' with fiber ''G'' where the structure group acts on the fiber by left multiplication. Since right multiplication by ''G'' on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by ''G'' on ''P''. The fibers of ''π'' then become right ''G''-torsors for this action.
 
The definitions above are for arbitrary topological spaces. One can also define principal ''G''-bundles in the [[category (mathematics)|category]] of [[smooth manifold]]s. Here ''π'' : ''P'' → ''X'' is required to be a [[smooth map]] between smooth manifolds, ''G'' is required to be a [[Lie group]], and the corresponding action on ''P'' should be smooth.
 
==Examples==
The prototypical example of a smooth principal bundle is the [[frame bundle]] of a smooth manifold ''M'', often denoted F''M'' or GL(''M''). Here the fiber over a point ''x'' in ''M'' is the set of all frames (i.e. ordered bases) for the [[tangent space]] ''T''<sub>''x''</sub>''M''. The [[general linear group]] GL(''n'','''R''') acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(''n'','''R''')-bundle over ''M''.
 
Variations on the above example include the [[orthonormal frame bundle]] of a [[Riemannian manifold]]. Here the frames are required to be [[orthonormal]] with respect to the [[metric tensor|metric]]. The structure group is the [[orthogonal group]] O(''n''). The example also works for bundles other than the tangent bundle; if ''E'' is any vector bundle of rank ''k'' over ''M'', then the bundle of frames of ''E'' is a principal GL(''k'','''R''')-bundle, sometimes denoted F(''E'').
 
A normal (regular) [[covering space]] ''p'' : ''C'' → ''X'' is a principal bundle where the structure group <math>\pi_1(X)/p_{*}(\pi_1(C))</math> acts on the fibres of ''p'' via the [[Covering space#Monodromy_action|monodromy action]]. In particular, the [[universal cover]] of ''X'' is a principal bundle over ''X'' with structure group <math>\pi_1(X)</math> (since the universal cover is simply connected and thus <math>\pi_1(C)</math> is trivial).
 
Let ''G'' be a Lie group and let ''H'' be a closed subgroup (not necessarily [[normal subgroup|normal]]). Then ''G'' is a principal ''H''-bundle over the (left) [[coset space]] ''G''/''H''. Here the action of ''H'' on ''G'' is just right multiplication. The fibers are the left cosets of ''H'' (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to ''H'').
 
Consider the projection π: ''S''<sup>1</sup> → ''S''<sup>1</sup> given by ''z'' ↦ ''z''<sup>2</sup>.  This principal '''Z'''<sub>2</sub>-bundle is the [[associated bundle]] of the [[Möbius strip]].  Besides the trivial bundle, this is the only principal '''Z'''<sub>2</sub>-bundle over ''S''<sup>1</sup>.
 
[[Projective space]]s provide some more interesting examples of principal bundles. Recall that the ''n''-[[sphere]] ''S''<sup>''n''</sup> is a two-fold covering space of [[real projective space]] '''RP'''<sup>''n''</sup>. The natural action of O(1) on ''S''<sup>''n''</sup> gives it the structure of a principal O(1)-bundle over '''RP'''<sup>''n''</sup>. Likewise, ''S''<sup>2''n''+1</sup> is a principal [[U(1)]]-bundle over [[complex projective space]] '''CP'''<sup>''n''</sup> and ''S''<sup>4''n''+3</sup> is a principal [[Sp(1)]]-bundle over [[quaternionic projective space]] '''HP'''<sup>''n''</sup>. We then have a series of principal bundles for each positive ''n'':
: <math>\mbox{O}(1) \to S(\mathbb{R}^{n+1}) \to \mathbb{RP}^n</math>
: <math>\mbox{U}(1) \to S(\mathbb{C}^{n+1}) \to \mathbb{CP}^n</math>
:<math>\mbox{Sp}(1) \to S(\mathbb{H}^{n+1}) \to \mathbb{HP}^n.</math>
Here ''S''(''V'') denotes the unit sphere in ''V'' (equipped with the Euclidean metric). For all of these examples the ''n'' = 1 cases give the so-called [[Hopf bundle]]s.
 
==Basic properties==
 
===Trivializations and cross sections===
 
One of the most important questions regarding any fiber bundle is whether or not it is [[trivial bundle|trivial]], ''i.e.'' isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:
 
:'''Proposition'''. ''A principal bundle is trivial if and only if it admits a global cross section.''
 
The same is not true for other fiber bundles. For instance, [[Vector bundle]]s always have a zero section whether they are trivial or not and [[fiber bundle#Sphere bundles|sphere bundles]] may admit many global sections without being trivial.
 
The same fact applies to local trivializations of principal bundles. Let ''π'' : ''P'' → ''X'' be a principal ''G''-bundle. An [[open set]] ''U'' in ''X'' admits a local trivialization if and only if there exists a local section on ''U''. Given a local trivialization <math>\Phi : \pi^{-1}(U) \to U \times G</math> one can define an associated local section <math>s : U \to \pi^{-1}(U)</math> by
:<math>s(x) = \Phi^{-1}(x,e)\,</math>
where ''e'' is the [[identity element|identity]] in ''G''. Conversely, given a section ''s'' one defines a trivialization Φ by
:<math>\Phi^{-1}(x,g) = s(x)\cdot g</math>
The simple transitivity of the  ''G'' action  on the fibers of ''P'' guarantees that this map is a [[bijection]], it is also a [[homeomorphism]]. The local trivializations defined by local sections are ''G''-[[equivariant]] in the following sense. If we write <math>\Phi : \pi^{-1}(U) \to U \times G</math> in the form <math>\Phi(p) = (\pi(p), \varphi(p))</math> then the map <math>\varphi : P \to G</math> satisfies
:<math>\varphi(p\cdot g) = \varphi(p)g.</math>
Equivariant trivializations therefore preserve the ''G''-torsor structure of the fibers. In terms of the associated local section ''s'' the map ''φ'' is given by
:<math>\varphi(s(x)\cdot g) = g.</math>
The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.
 
Given an equivariant local trivialization ({''U''<sub>''i''</sub>}, {Φ<sub>''i''</sub>}) of ''P'', we have local sections ''s''<sub>''i''</sub> on each ''U''<sub>''i''</sub>. On overlaps these must be related by the action of the structure group ''G''. In fact, the relationship is provided by the [[transition function]]s
:<math>t_{ij} = U_i \cap U_j \to G\,</math>
For any ''x'' in ''U''<sub>''i''</sub> ∩ ''U''<sub>''j''</sub> we have
:<math>s_j(x) = s_i(x)\cdot t_{ij}(x).</math>
 
===Characterization of smooth principal bundles===
 
If ''π'' : ''P'' → ''X'' is a smooth principal ''G''-bundle then ''G'' acts freely and [[proper map|properly]] on ''P'' so that the orbit space ''P''/''G'' is [[diffeomorphic]] to the base space ''X''. It turns out that these properties completely characterize smooth principal bundles. That is, if ''P'' is a smooth manifold, ''G'' a Lie group and ''μ'' : ''P'' × ''G'' → ''P'' a smooth, free, and proper right action then
*''P''/''G'' is a smooth manifold,
*the natural projection ''π'' : ''P'' → ''P''/''G'' is a smooth [[submersion (mathematics)|submersion]], and
*''P'' is a smooth principal ''G''-bundle over ''P''/''G''.
 
==Use of the notion==
 
===Reduction of the structure group===
{{see also|Reduction of the structure group}}
Given a subgroup ''H'' of ''G'' one may consider the  bundle <math>P/H</math> whose fibers are homeomorphic to the [[coset space]] <math>G/H</math>.  If the new bundle admits a global section, then one says that the section is a '''reduction of the structure group from ''G'' to ''H'' '''.  The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of ''P'' which is a principal ''H''-bundle. If ''H'' is the identity, then a section of ''P'' itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.
 
Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal ''G''-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from ''G'' to ''H''). For example:
 
* A 2''n''-dimensional real manifold admits an [[almost-complex structure]] if the [[frame bundle]] on the manifold, whose fibers are <math>GL(2n,\mathbb{R})</math>, can be reduced to the group <math>\mathrm{GL}(n,\mathbb{C}) \subset \mathrm{GL}(2n,\mathbb{R})</math>.
 
* An ''n''-dimensional real manifold admits a ''k''-plane field if the frame bundle can be reduced to the structure group <math>\mathrm{GL}(k,\mathbb{R}) \subset \mathrm{GL}(n,\mathbb{R})</math>.
 
* A manifold is [[orientable]] if and only if its frame bundle can be reduced to the [[special orthogonal group]], <math>\mathrm{SO}(n) \subset \mathrm{GL}(n,\mathbb{R})</math>.
 
* A manifold has [[spin structure]] if and only if its frame bundle can be further reduced from <math>\mathrm{SO}(n)</math> to <math>\mathrm{Spin}(n)</math> the [[Spin group]], which maps to <math>\mathrm{SO}(n)</math> as a double cover.
 
Also note: an ''n''-dimensional manifold admits ''n'' vector fields that are linearly independent at each point if and only if its [[frame bundle]] admits a global section. In this case, the manifold is called [[parallelizable]].
 
===Associated vector bundles and frames===
 
{{See also| Frame bundle}}
If ''P'' is a principal ''G''-bundle and ''V'' is a [[linear representation]] of ''G'', then one can construct a vector bundle <math>E=P\times_G V</math> with fibre ''V'', as the quotient of the product ''P''&times;''V'' by the diagonal action of ''G''. This is a special case of the [[associated bundle]] construction, and ''E'' is called an [[associated vector bundle]] to ''P''. If the representation of ''G'' on ''V'' is [[faithful representation|faithful]], so that ''G'' is a subgroup of the general linear group GL(''V''), then ''E'' is a ''G''-bundle and ''P'' provides a reduction of structure group of the frame bundle of ''E'' from GL(''V'') to ''G''. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.
 
==Classification of principal bundles==
 
{{Main| Classifying space}}
Any topological group ''G'' admits a '''classifying space''' ''BG'': the quotient by the action  of ''G'' of some [[weakly contractible]] space ''EG'', ''i.e.'' a topological space with vanishing [[homotopy group]]s. The classifying space has the property that any ''G'' principal bundle over a [[paracompact]] manifold ''B'' is isomorphic to a [[pullback bundle|pullback]] of the principal bundle <math>EG\longrightarrow BG.</math>.<ref>{{Citation | last1=Stasheff | first1=James D. | title=Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | year=1971 | chapter=''H''-spaces and classifying spaces: foundations and recent developments | pages=247–272}}, Theorem 2</ref> In fact, more is true, as the set of isomorphism classes of principal ''G'' bundles over the base ''B'' identifies with the set of homotopy classes of maps ''B'' → ''BG''.
 
==See also==
*[[Associated bundle]]
*[[Vector bundle]]
*[[G-structure]]
*[[Reduction of the structure group]]
*[[Gauge theory]]
*[[Connection (principal bundle)]]
 
==References==
<references/>
 
==Books==
*{{cite book | first = David | last = Bleecker | title = Gauge Theory and Variational Principles | year = 1981 | publisher = Addison-Wesley Publishing | isbn = 0-486-44546-1 (Dover edition)}}
*{{cite book | first = Jürgen | last = Jost | title = Riemannian Geometry and Geometric Analysis | year = 2005 | edition = (4th ed.) | publisher = Springer | location = New York | isbn = 3-540-25907-4}}
*{{cite book | last = Husemoller | first = Dale | title = Fibre Bundles | publisher = Springer | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}}
*{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year = 1997 | isbn = 0-387-94732-9}}
*{{cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | publisher = Princeton University Press | location = Princeton | year = 1951 | isbn = 0-691-00548-6}}
 
{{DEFAULTSORT:Principal Bundle}}
[[Category:Fiber bundles]]
[[Category:Differential geometry]]
[[Category:Group actions]]

Revision as of 05:17, 13 January 2014

In mathematics, a principal bundle[1][2][3][4] is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with

  1. An action of G on P, analogous to (x,g)h = (x, gh) for a product space.
  2. A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) → x.

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X × G → G which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.

A common example of a principal bundle is the frame bundle FE of a vector bundle E, which consists of all ordered bases of the vector space attached to each point. The group G in this case is the general linear group, which acts in the usualTemplate:Clarify way on ordered bases. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.

Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.

Formal definition

A principal G-bundle, where G denotes any topological group, is a fiber bundle π : P → X together with a continuous right action P × G → P such that G preserves the fibers of P (i.e. if y ∈ Px then yg ∈ Px for all g ∈ G) and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group G itself. Frequently, one requires the base space X to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of π : P → X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P/G is homeomorphic to the base space X. Because the action is free, the fibers have the structure of G-torsors. A G-torsor is a space which is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal G-bundle is as a G-bundle π : P → X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of π then become right G-torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal G-bundles in the category of smooth manifolds. Here π : P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth.

Examples

The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold M, often denoted FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space TxM. The general linear group GL(n,R) acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(n,R)-bundle over M.

Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group O(n). The example also works for bundles other than the tangent bundle; if E is any vector bundle of rank k over M, then the bundle of frames of E is a principal GL(k,R)-bundle, sometimes denoted F(E).

A normal (regular) covering space p : C → X is a principal bundle where the structure group π1(X)/p*(π1(C)) acts on the fibres of p via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group π1(X) (since the universal cover is simply connected and thus π1(C) is trivial).

Let G be a Lie group and let H be a closed subgroup (not necessarily normal). Then G is a principal H-bundle over the (left) coset space G/H. Here the action of H on G is just right multiplication. The fibers are the left cosets of H (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to H).

Consider the projection π: S1 → S1 given by z ↦ z2. This principal Z2-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal Z2-bundle over S1.

Projective spaces provide some more interesting examples of principal bundles. Recall that the n-sphere Sn is a two-fold covering space of real projective space RPn. The natural action of O(1) on Sn gives it the structure of a principal O(1)-bundle over RPn. Likewise, S2n+1 is a principal U(1)-bundle over complex projective space CPn and S4n+3 is a principal Sp(1)-bundle over quaternionic projective space HPn. We then have a series of principal bundles for each positive n:

O(1)→S(ℝn+1)→ℝℙn
U(1)→S(ℂn+1)→ℂℙn
Sp(1)→S(ℍn+1)→ℍℙn.

Here S(V) denotes the unit sphere in V (equipped with the Euclidean metric). For all of these examples the n = 1 cases give the so-called Hopf bundles.

Basic properties

Trivializations and cross sections

One of the most important questions regarding any fiber bundle is whether or not it is trivial, i.e. isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

Proposition. A principal bundle is trivial if and only if it admits a global cross section.

The same is not true for other fiber bundles. For instance, Vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let π : P → X be a principal G-bundle. An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization Φ:π−1(U)→U×G one can define an associated local section s:U→π−1(U) by

s(x)=Φ−1(x,e)

where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by

Φ−1(x,g)=s(x)⋅g

The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection, it is also a homeomorphism. The local trivializations defined by local sections are G-equivariant in the following sense. If we write Φ:π−1(U)→U×G in the form Φ(p)=(π(p),φ(p)) then the map φ:P→G satisfies

φ(p⋅g)=φ(p)g.

Equivariant trivializations therefore preserve the G-torsor structure of the fibers. In terms of the associated local section s the map φ is given by

φ(s(x)⋅g)=g.

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization ({Ui}, {Φi}) of P, we have local sections si on each Ui. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions

tij=Ui∩Uj→G

For any x in Ui ∩ Uj we have

sj(x)=si(x)⋅tij(x).

Characterization of smooth principal bundles

If π : P → X is a smooth principal G-bundle then G acts freely and properly on P so that the orbit space P/G is diffeomorphic to the base space X. It turns out that these properties completely characterize smooth principal bundles. That is, if P is a smooth manifold, G a Lie group and μ : P × G → P a smooth, free, and proper right action then

  • P/G is a smooth manifold,
  • the natural projection π : P → P/G is a smooth submersion, and
  • P is a smooth principal G-bundle over P/G.

Use of the notion

Reduction of the structure group

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Given a subgroup H of G one may consider the bundle P/H whose fibers are homeomorphic to the coset space G/H. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of P which is a principal H-bundle. If H is the identity, then a section of P itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal G-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from G to H). For example:

  • An n-dimensional real manifold admits a k-plane field if the frame bundle can be reduced to the structure group GL(k,ℝ)⊂GL(n,ℝ).

Also note: an n-dimensional manifold admits n vector fields that are linearly independent at each point if and only if its frame bundle admits a global section. In this case, the manifold is called parallelizable.

Associated vector bundles and frames

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Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. If P is a principal G-bundle and V is a linear representation of G, then one can construct a vector bundle E=P×GV with fibre V, as the quotient of the product P×V by the diagonal action of G. This is a special case of the associated bundle construction, and E is called an associated vector bundle to P. If the representation of G on V is faithful, so that G is a subgroup of the general linear group GL(V), then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL(V) to G. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.

Classification of principal bundles

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Any topological group G admits a classifying space BG: the quotient by the action of G of some weakly contractible space EG, i.e. a topological space with vanishing homotopy groups. The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback of the principal bundle EG⟶BG..[5] In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.

See also

References

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 page 35
  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 page 42
  3. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 page 37
  4. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 page 370
  5. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Theorem 2

Books

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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