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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Effect of gauge transformation on connection 1-form of a principal connection

Let $(P,\pi,M,G)$ be a principal fibration, $A$ a principal connection on $P$ (i.e. $\forall p \in P, T_pP = A_p \oplus V_p$), $\omega$ the connection 1-form of $A$, $f$ a gauge transformation of $P$, ...
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2answers
45 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
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3answers
713 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then $\...
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1answer
25 views

Preimage of principal bundle under equivariant map

Let $M$ be a manifold, $G,H$ be some Lie groups, $\sigma:G\to H$ be a Lie group homomorphism, $K\subset H$ a maximal compact subgroup of $H$ and $\tilde K:=\sigma^{-1}(K)\subset G$ . Let further $\...
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62 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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1answer
64 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
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1answer
15 views

Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
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10 views

local sections transformation formula

I am trying to prove a formula which states the relations between two local sections in a principal bundle: Let $P(M,G)$ be a principal bundle let $\{ U_\alpha \}_{\alpha \in I}$ be an open cover for ...
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1answer
67 views

Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
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21 views

Holonomy of a curve in case of principal $U(1)$ bundle

Suppose $\pi : P\rightarrow M$ is principal $U(1)$ bundle. Let $\gamma$ be a loop in $M$ based at $x_0$ and write $iA$ as connection 1-form on $P$ where $A\in \Omega(P)$. Now define $hol_{\gamma}(A)\...
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Sign disagreement in curvature form on $U(1)$-bundle

this is rather trivial question to masters, but as I'm totally disoriented by my computation, so deciding to ask. I'm trying to solve the following exercise, 1.1, showing the curvature 2-form $F(X,Y)=...
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1answer
28 views

Apply connections to a gauge transformation?

I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $...
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1answer
51 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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37 views

Making a principal bundle into a covering space

Suppose $\pi : P\rightarrow M$ is a principal $G$-bundle. I want to make this into a covering map by changing the topology of $P$. By local triviality we can find for each $x\in M$ an open $U\subset ...
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18 views

Definition of isomorphism classes of vector bundles with reduced structure group

I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq ...
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34 views

Möbius Band Bundle $(Mo,\mathbb{S}^1,\text{proj}_1,\mathbb{R}) $ is not a Principal $\mathbb{R}$-bundle

This is claimed in various places. The problem seems to be with finding a free and transitive group action that has the fibers of $Mo$ as its orbits. I construct $Mo$ as $$ Mo = \mathbb{S}^1 \times \...
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26 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 ...
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1answer
26 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
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1answer
23 views

The bundle of spin frames as an associated bundle

$\DeclareMathOperator{\Spin}{Spin}$Let $X$ be an oriented smooth $n$-manifold with the frame bundle $\pi_{SO} \colon F_{SO} \to X$. Then the bundle of spin frames is a $\Spin(n)$-bundle $\pi_{\Spin} \...
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1answer
23 views

Effect of mapping of principal fiber bundles on a principal connection

Let $\lambda = (P,\pi,M,G)$ and $\lambda' = (P',\pi',M',G')$ be two principal fiber bundles, $\lambda$ being equiped with a principal connexion whose connexion form is $\omega$. If $(f,k,\rho)$ is a ...
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149 views

Does every manifold M always admit a Riemannian metric?

In the book "Geometry and Topology for Physicists" by Nash and Sen, in Section 7.6, after showing that the structure group $GL(n,\mathbb{R})$ of a frame bundle $F(M)$ (for a general manifold $M$ of ...
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1answer
25 views

why does the regular action of the structure group not imply triviality of a fibre system?

Let $P$ and $X$ be algebraic varieties, $\pi:P\to X$ a morphism and $G$ an algebraic group acting on $P$. Serre calls the triple $(G,P,X)$ a fibre system an proves that if it is locally isotrivial (i....
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29 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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317 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
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29 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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1answer
57 views

Confused by two different perspectives on $G$-vector bundles

I'm trying to understand how these two perspectives on vector bundle with a $G$-action come together. Perspective 1: Let $P \to X$ be a principal $G$-bundle. The associated bundle construction gives ...
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23 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 ...
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1answer
63 views

What are classifying spaces actually classifying?

Let $G$ be a group. When we say the classifying space of $G$ we are actually meaning the classifying space of the principal $G-$bundles because the notion of classifying spaces is about classifying ...
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2answers
66 views

Moduli space of stable principal $G$-bundles on a compact Riemann surface.

Let $C$ be a compact Riemann surface. I'm looking for some references in order to try to understand what is the moduli space of stable principal $G$-bundles on $C$, where $G$ is a simple Lie group. ...
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3answers
89 views

Flat non-trivial $U(1)$-bundle? Is it possible?

maybe this is a very stupid question and I'm missing something very trivial. It's well known that $U(1)$-bundles are classified by the Euler class or the first Chern class. More precisely, the ...
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1answer
109 views

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles on differential manifolds, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of ...
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1answer
48 views

Explicit Classifying map for a vector bundle

I'm interested in writing down explicitly the classifying map for a given real vector bundle $\pi\colon E \to B$ of rank, say $k$. Take $B$ to be a compact manifold (I'm interested in this case) so we ...
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1answer
29 views

Local one sections and connections $1$-forms?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle endowed with a connection $1$-form $\omega\in \Omega^1(P; \mathfrak{g})$ where $\mathfrak{g}$ is the Lie algebra of $G$. Let $\{U_i\}_{i\in I}$ ...
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1answer
85 views

Is there a nontrivial fiber or principal bundle over $S^3$?

Is there a nontrivial fiber or principal bundle over $S^3$?I know that, by a paper of Steenrod,see the link below, every sphere bundle on 3- sphere is trivial but what about arbitrary fiber ...
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2answers
72 views

Defining a differentiable structure by means of functions.

I am trying to understand the construction of principal bundles from Kobayashi and Nomizu, and the situation is the following. Let $M$ be a manifold, $\{ U_\alpha \}_{\alpha \in A}$ an open covering ...
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1answer
27 views

Are there multiple non-isomorphic principal $G$-bundles on Euclidean space? [duplicate]

I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :). I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't ...
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81 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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1answer
34 views

What does $P\times_G V\to B$ mean?

Let $$\pi:P\to B$$ be a principal $G$-bundle and $$\rho:G\times V\to V$$ a continuous action of $G$ on the vector space $V$. What does the notation $P\times_G V\to B$ mean? It is supposed to be ...
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1answer
88 views

Interpretation for the curvature and monodromy of a connection - Reality check

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on: A connection is flat (vanishing curvature)...
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2answers
33 views

Parallel displacement on principal bundles

Let $\pi : P \to M$ be a principal bundle with structure group $G$ and connection $\Gamma$. For a fixed $x \in M$, denote by $\Omega(x)$ the space of piecewise differentiable loops based at $x$. Every ...
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2answers
34 views

Lift of a curve in a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle and $I:=[0, 1]$. Given a curve $\alpha:I\longrightarrow M$ and $p_0\in \pi^{-1}(\alpha(0))$ how can I show there is a curve $\beta:I\...
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25 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 $P$,...
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1answer
50 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then $H_p:=\...
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1answer
31 views

Isomorphism of Lie algebras $\mathfrak{g}\simeq \mathfrak{X}^v(P)$?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. For $p\in P$, $V_p$ denotes the space of tangent vertical vectors, that is, $V_p:=T_pP_{\pi(p)}$. The space of vertical vector fields will be ...
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345 views

Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of problem?...
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33 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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1answer
41 views

Classification of $G$-principal Bundle and classification of $G$-coverings: a bridge between the two?

I encountered the following sentence in an exercise (the context is irrelevant) Let $G\cong\langle s_1,s_2,\dots , s_g \mid R \rangle$ be a discrete one relator group. Consider the $G$-principal ...
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34 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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60 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 $Bun(X)$...
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1answer
58 views

Principal bundles with compact simply connected structure group over 2-manifolds

I'm reading Thomas Friedrich's "Dirac Operators in Riemannian Geometry," where the following is stated (in the Remark on page 42 before section 2.2 begins, if anyone is following along with the ...