In mathematics, a principal bundle 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 $G$.

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Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
6
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249 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
5
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107 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
5
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91 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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58 views

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
5
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54 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
4
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57 views

Classifying vector bundles with a reduction of its structure group

Let $Bun(X)$ denote the set of equivalence classes of complex rank 2 vector bundles with a reduction of its structure group to $\mathbb{H}^*$. How can I proof that there is a bijection between ...
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87 views

Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
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37 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy ...
4
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153 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
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70 views

Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a 3 -manifold which may be viewed as a circle bundle over ...
3
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23 views

Two ways to define the fundamental vector field on a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. Let $\mathfrak{g}$ be the Lie algebra of left-invariant vector fields on $G$ and $\mathfrak{V}(P)$ be the space of vertical vector fields on ...
3
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57 views

Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...
3
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69 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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150 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
2
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21 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
2
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30 views

Spin^c structure induced by a spin structure

I wondered how it works exactly to induce a $\mathrm{spin^c}$-structure if a spin structure is given. I wanted to use the following definitions as used in Friedrich`s "Dirac operators in Riemannian ...
2
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34 views

Reduction to the special orthogonal group

It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle However, I can't see why, because the isomorphism $E\cong E^*$ means ...
2
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45 views

How to Induce a Metric on Homogeneous Space $G/H$ by the Metric from Bundle G

I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G. More specifically and simply, consider principal ...
2
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74 views

Conjugate group homomorphisms induce homotopic maps on classifying spaces

Let $G$ be a group and let $\phi: G \to G$ be the inner automorphism given by conjugation by an element $g' \in G$, i.e., $\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ...
2
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41 views

Identity of curvature in a principal bundle

I saw in wikipedia (http://en.m.wikipedia.org/wiki/Curvature_form) the following identity for the curvature 2-form of a principal bundle $$2\Omega(X,Y)=h[X,Y]-[hX,hY]$$ where $X,Y\in T_uP$, $P$ being ...
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63 views

What is the principal bundle structure of $O(n)$?

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question ...
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41 views

Contraction of second exterior covariant derivative with metric

Let $G \hookrightarrow P \to M$ be a principal $G$ bundle, $P \times_\rho V$ be a vector bundle associated to representation $\rho$ of $G$ on $V$. If $\omega$ is a connection $1$-form on $P$ then we ...
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49 views

Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
2
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64 views

How to decompose connections on the complexified orthonormal frame bundle?

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} ...
2
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69 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
2
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174 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} ...
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45 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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27 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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25 views

Sectional category (Schwarz genus) of the Milnor join construction

Assume topological spaces to be normal and paracompact. Following the article: "The genus of a fiber space" by A. Schwarz, we call the sectional category (or Schwarz genus) of a locally trivial fiber ...
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74 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
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31 views

Description of complex conjugate $\operatorname{Spin}^c$ structure without cocycles

The following uses exclusively cocycle descriptions for spin and spinc structures which I would like to avoid. See for example Nicolaescu "Notes on Seiberg-Witten invariants", pages 40-41 for their ...
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31 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
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13 views

$G$-invariant vector field coming from a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. If $(U, \phi)$ is a local trivialization of this bundle then for every $x\in M$ we have a diffeomorphism $$\phi_x:P_{x}\longrightarrow G, ...
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33 views

Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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41 views

Bundle map is isomorphism iff it covers a homeomorphism

Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ ...
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37 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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77 views

Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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0answers
51 views

Certain principle bundle structure on $\mathbb{R}^{n}\setminus \{0\}$

Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a principle fibre bundle. Here $\mathbb{H}^{2}$ is the Poincare upper plane ...
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27 views

How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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30 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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27 views

extension of a principal connection

I am trying to prove the following: Suppose that $\alpha:H\to G$ is a Lie group homomorphism and let $P\to M$ be a principal $H$-bundle and $Q\to M$ a principal $G$-bundle. Suppose further that there ...
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98 views

Relation between principal bundle automorphisms and maps in $\Lambda^0(M,Ad(P))$

Let $ G \hookrightarrow P \xrightarrow{ \pi } M $ be a principal $ G $-bundle over $M$. Denote by $\mathrm{Ad} (P) = P \times _{ \mathrm{Ad} } G $ the non-linear adjoint bundle. It seems to me that ...
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0answers
43 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
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82 views

morphisms of principal bundles with different structure groups

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram: \begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G ...
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0answers
98 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
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31 views

Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
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44 views

$\mathbb{RP}^3 \rightarrow \mathbb{CP}^1$ defines a principal U(1)-bundle

I have to show that the map $\pi: (x_o : x_1 : x_2 : x_3) \in \mathbb{RP}^3 \rightarrow (x_0 + i x_1) : (x_2+ ix_3) \in \mathbb{CP}^1$ defines a principal U(1)-bundle. The two standard coordinate ...
0
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0answers
13 views

Principal bundle isomorphism.

Let $G\longrightarrow P\overset{\pi}{\longrightarrow} M$ be a differentiable principal bundle, i.e. $M$ and $P$ are differentiable manifolds, $G$ is a Lie group, $\pi$ is a differentiable surjective ...
0
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25 views

induced connections

I am studying "Spin Geometry" and i have the following question. In page 106 after the connection 1-form is defined there is a discussion about induced connections.Supposing we have a connection on ...