Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [principal-bundles]

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 $G$.

0
votes
0answers
13 views

Is the curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the $\textit{exterior ...
0
votes
0answers
21 views

Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $\omega$ on a principal bundle and its curvature form $\Omega$, I can state that $\Omega=d_\omega\omega$ alike I do in the ...
1
vote
0answers
18 views

Classifying maps and transition functions

Suppose a space $X'$ is obtained from $X$ by attaching an $i$-cell, i.e., $$X' = X \cup_\phi e^i$$ where $\phi: \partial e^i \to X$ is the attaching map. Let $G$ be a structure group, and $P$ be a ...
0
votes
0answers
16 views

Change of trivialization

Let $\pi:P\to M$ a smooth $G$-principal bundle with action $\beta:G\times P\to P$, $G$ a Lie group and $M$ a diff. manifold. Let also $T^*P$ be its cotangent bundle and $\sigma$ be a section of this ...
1
vote
0answers
15 views

Principal fibre bundles with constant transition functions

I guess this is not limited to principal bundles, however those are my primary interest in asking this question. Let $(P,\pi,M,G)$ be a principal fibre bundle over $M$ with structure group $G$. ...
0
votes
2answers
56 views

Non-flat connection on trivial bundle?

From what I have read it seems like there exists non-flat connections on trivial vector/principal bundles. However I can't find any notes on it or examples. Can anyone confirm that such connections ...
6
votes
1answer
49 views

An almost complex structure on $M$ is equivalent to a reduction of the structure group of the tangent bundle

Let $M$ be an $2n$-dimensional manifold. Let $\mathcal{F}_{\mathrm{GL}(2n, \mathbb{R})}$ be the frame bundle over $M$. Consider the subgroup $\mathrm{GL}(n, \mathbb{C})\subset\mathrm{GL}(2n, \mathbb{R}...
1
vote
0answers
22 views

Different universal bundle definitions

When reading about classifying spaces and universal (principal) bundles, multiple definitions seem to occur. One has the following possibilities (as far as I can see): The universal $G$-bundle $EG\...
0
votes
0answers
22 views

parallel transport is independent of the bases chosen in each of the two tangent spaces

Connections on principal fibre bundles In the above set of notes on page-3 section 2.2 under the heading Parallel transport there is a statement that equivariance ensures that the parallel transport ...
4
votes
0answers
65 views

Problem with equivalent definition of a integrable $G$-structure

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition: My problem is that I don't understand the converse of this ...
1
vote
1answer
30 views

When is a quotient of a principal bundle is a principal bundle?

Suppose $\pi:P\rightarrow M$ is a principal $G$ bundle. Let $H$ be a Lie group acting freely and properly on $P$ and on $M$ so that $P/H$ and $M/H$ are manifolds. Further assume this action is such ...
0
votes
1answer
43 views

Question about group of automorphism of some $G$-structure.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15. I don't understand why $U$ consists of transformation $a$ of $M$ that leave each ...
3
votes
1answer
54 views

Deducing the statement $I(G) \rightarrow H^*(BG; \Bbb R)$, Chern-weil theory

So I know: Let $G$ be a Lie group Given a smooth principal $G$ bundle $P \rightarrow M$, we may define an algebra homomorphism $$I(G) \rightarrow H^{ev}(M; \Bbb R)$$ where $I(G)$ the graded ...
0
votes
1answer
29 views

Universal principal bundle of a subgroup $H \subseteq G$.

This was claimed in the referenced link: If $H \subseteq G$ is an admissible subgroup, i.e. $G \rightarrow G/H$ is a principal $H$ bundle. Let $EG \rightarrow BG$ be a model universal ...
6
votes
0answers
28 views

Characteristic classes for $P \rightarrow G \rightarrow G/P$

Let $G$ be a complex semisimple Lie group and let $P$ be a parabolic subgroup. We know that the cohomology of the flag variety $G/P$ is generated by Schubert classes. There is a principal $P$ bundle, ...
0
votes
1answer
28 views

Help find mistake in conclusion about vector fields on principal bundle

Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of ...
0
votes
0answers
26 views

Homotopy invariance of pullbacks of principal bundles

This is the proof of lemma 7.2 in a notes by Stephen Mitchell, on classifying spaces. Essentially one step of the proof claims that: Let $p:Y \rightarrow B \times I$ be a principal bundle. If $B$...
3
votes
1answer
73 views

Literature suggestion for understanding Gauge theory from the perspective of a Mathematician.

Can anyone please suggest some good literature or references for understanding Gauge theory from the perspective of a mathematician (from the point of view of differential geometry)? Being a ...
1
vote
0answers
121 views

Exact sequence induced by an exact sequence of groups

Suppose $H\hookrightarrow G$ is an inclusion of topological groups and $H$ is closed in $G$. Then we have an exact sequence $$ 1\to H\to G\xrightarrow \pi G/H \to 1$$ where $G/H$ is just a pointed ...
1
vote
1answer
43 views

Computing the differential of a Lie group action

(Ex. 27.4 page 252 Loring Tu) (The differential of an action). Let $\mu: P \times G \rightarrow P$. For $g \in G$, the tangent space $T_gG$ may be identified with $l_{g*} \mathfrak{g} $, where $...
4
votes
1answer
53 views

Does maps between fundamental groups induces a continuous map between spaces?

Main question is this: Suppos $M$ is a manifold and $G$ is a finite group. If there is a group homomorphism $\phi:\pi_1 M\to G$, is there a continuous map $f:M\to BG$, where $BG$ is a classifying ...
0
votes
0answers
25 views

Natural bijection between equivariant maps and sections, Principal bundles

This is in page 11, Prop 6.1 of Mitchell's notes on Fibre bundles. I am quoting the proposition below. Let $\pi:P \rightarrow B$ be a principal $G$ bundle, $X$ a right $G$-space. $Hom_G(P,X)$ ...
2
votes
2answers
147 views

Doubt in the definition of principal bundle.

I am following Kobayashi and Nomizu( Foundations of differential geometry) Volume 1. In page number 50 while defining principle $G$-bundle $P(M,G)$ they said that the action of the Lie group $G$ ...
0
votes
1answer
40 views

Question about a proposition in Kobayashi book about $G$-structures.

I'm reading Kobayashi's book, Transformation Groups in Differential Geometry and at the page 3 is this proposition: the definition of $K$ is given here: My question is why this proposition is ...
1
vote
1answer
71 views

Reduced bundles and global sections of associated bundle

I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6. The structure group of a principal bundle $P(M,G)$ is reducible to a closed ...
0
votes
0answers
23 views

Local frame inducing a map of principal bundles

Let $V \rightarrow M$ a vector bundle. $P \rightarrow M$ a principal $G$-bundle. Let $\phi:G \rightarrow GL(V)$ be a representation. A local section $s$ for $P$, frame bundle for $V \rightarrow M$ , ...
0
votes
0answers
27 views

$H$ is admissible, then $P\rightarrow P/H$ is a principle $H$ -bundle.

Proposition 3.5, page 5: Suppose $P \rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P \rightarrow P/H$ is a principal $H$-bundle. ...
1
vote
1answer
55 views

Different notions of torsors in algebraic geometry

In what follows $X$ will be a scheme and $G$ a group scheme. In the examples I will take $X=\mathbb{P}^1_k$ and $G=\mathbb{G}_{m}$. When reading about "the torsor..." I found many definitions, not ...
0
votes
0answers
35 views

Connection $1$-form acting on vector fields

I'm reading this paper about the c-map between special Kähler manifolds and Hyperkähler manifolds and in the introduction the authors talk about the cotangent bundle as a certain associated bundle of ...
0
votes
0answers
33 views

Connection/Curvature as a matrix of Real valued forms

Let $P(M,G)$ be a principal $G$ bundle. Let $\omega$ be a connection $1$ form on $P(M,G)$. This is a $\mathfrak{g}$ valued $1$ form on $P$ i.e., for each $p\in P$, we have $\omega(p):T_pP\rightarrow ...
2
votes
2answers
48 views

What does determinant bundle of a principal bundle say about the principal bundle

Let $\pi:P\rightarrow M$ be a principal $Gl(n,\mathbb{R})$ bundle. Given $x\in M$ there is an open set $U$ containing $x$ and a local trivialization $\pi^{-1}(U)\rightarrow U\times G$. This gives a ...
2
votes
1answer
32 views

Isotropy group of connection is isomorphic to centraliser of holonomy group

I am asking for a proof of Lemma (4.2.8) of Donaldson, Kronheimer: The Geometry of Four-Manifolds. Let $P \rightarrow X$ be a principal bundle with structure group $G$. Denote by $\mathcal{G}$ the ...
2
votes
1answer
37 views

Is there a group whose manifold is a fiber bundle with base is $S_1$ and fiber $\mathbb{Z_2}?$

Let's consider a fiber bundle with base $S_1$ and fiber $\mathbb{Z}_2$. I want this manifold to be topologically non-trivial, the edge of the Möbius strip. How do I know if is it possible to ...
6
votes
0answers
142 views

Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
2
votes
1answer
95 views

Showing $GL_n$ is a special algebraic group

So there's this notion of a group scheme $G$ being 'special' if any principal $G$-bundle over a scheme $X$ (say defined in the etale topology) is also locally trivial in the Zariski topology. I would ...
0
votes
0answers
9 views

Universal standard principle bundle of the Gauge group

Let $p:P\rightarrow B$ be a principal $G$-bundle, and let $E_G\rightarrow B_G$ be the universal bundle for $G$. Let $Aut_B(P)$ be the subspace of $Map_G(P,P)$ of maps over $B$. Let $Map_P(B,B_G)$ be ...
0
votes
1answer
26 views

Problem about frame bundle in Kobayashi's book

I'm reading Kobayashi's book "Transformation Groups in Differential Geometry" and I have a problem in the proof of this lemma: At the converse part he says this: My question is about $f,$ namely, ...
1
vote
1answer
48 views

Principal bundles with quotient map

I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $\pi: G \rightarrow G/H$, then $(G, \pi, G/H, H)$ is a $H$-principal bundle ...
0
votes
1answer
21 views

Quick question on structure groups of the frame bundle of a Manifold

Let $M$ a manifold and $G$ a group. Is it true that the statement: "the structure group of the frame bundle of $M$ can be reduced to $G$" simply means: There is a subbundle $S$ of the frame ...
4
votes
0answers
42 views

Extend the fiber of a principal $PSU(n)$-bundle

For $n>2$, the outer automorphism group of $PSU(n)$ is $\mathbb{Z}_2$. My question: Given a principal $PSU(n)$-bundle $P$ over a manifold $M$, can we extend the fiber of $P$ to $\mathbb{Z}_2\...
0
votes
0answers
17 views

Any example of optimization on a fiber bundle?

In engineering, concepts like manifolds (with boundary) arise naturally in constrained optimization. Sometimes, the domain of optimization is naturally identified with a Lie group. Theoretically ...
1
vote
0answers
32 views

Defining a connection in $\mathbb{R}^2$ using a connection $1$-form

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality). In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb{R}^4$ and a ...
0
votes
0answers
21 views

Fibred charts adapted to principal bundle structures

If $\pi_E:E\rightarrow M$ is a rank $k$ vector bundle (let's assume everything in this question to be real for simplicity), it is the most common to use fibred charts adapted to the vector bundle as ...
0
votes
0answers
23 views

Principal Bundle definition

Let $G$ act on smooth manifold $M$. If the action is free $Stab(G) = \{ e\}$, then $O(p) = Orbit(p) \approx G$. In the definition of principal bundle with $(E,\pi, B, G)$, then let $G$ acting on $E$ ...
0
votes
0answers
27 views

Associativity of balanced products for $G$-spaces

This is Proposition 3.1 pg 4 Let $X$ be a right $G$-space, $Y$ a $(G,H)$ space, $Z$ a left $H$ space, then there is a natural homeomorphism. $$(X \times _G Y) \times_H Z \cong X \times_G( Y \...
1
vote
1answer
46 views

principal $G$ bundle from principal $H$ bundle given a morphism of Lie groups $\phi:G\rightarrow H$

Let $\phi:G\rightarrow H$ be a morphism of Lie groups. Given a principal $G$ bundle, we can associate a principal $H$ bundle by what is called associated fiber bundle for a principal bundle. Can we ...
0
votes
1answer
31 views

changing the structure group along the given homomorphism of Lie groups

Let $P\rightarrow M$ be a principal $G$ bundle and $\phi:H\rightarrow G$ be a morphism of Lie groups. Can some one help me to understand the constrution of reducing the structrue group to $H$ i.e., ...
0
votes
1answer
42 views

Map from schemes to stacks

I have just started studying stacks. Trying to understand the theory I was thinking about a (very interesting) toy example: $ BG $ the classifying stack of a smooth (over a base scheme $ S $) group G. ...
0
votes
0answers
38 views

Transition maps of principal bundle are smooth

Let $P(M,G)$ be a principal bundle. We choose an open covering $\{U_\alpha\}$ of $M$ and trivializations $\psi_\alpha:\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times G$ defined as $\psi_\alpha(u)= (\...
2
votes
1answer
65 views

Looking at the connection 1-form on a principal G-bundle in coordinates

I'm reading this paper, and I'm confused about something. Let $A$ be a connection on a principal $G$-bundle $P$ over $\mathbb{R}^4$, and $F(A)$ its curvature. Let $\mathrm{ad}(P)=P\times_G\mathfrak{g}...