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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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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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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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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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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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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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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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 ...
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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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282 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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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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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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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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57 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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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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81 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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106 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
28 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
78 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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630 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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71 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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25 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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75 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
33 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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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 ...
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31 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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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 ...
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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 ...
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1answer
49 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 ...
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30 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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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 ...
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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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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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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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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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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 ...
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Isomorphism between tangentspace of Lie group quotient and quotient of Lie algebras.

Let $G$ be a Lie group and $H$ a closed subgroup with Lie algebras $\mathfrak g, \mathfrak h$. Then the canonical projection $p: G \to G/H$ is a submersion. Fix a $g \in G$. We have a linear ...
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Gauge transformations for line bundle where the manifold is simply connected.

Im trying to understand the significance of the manifold being simply connected for the following (or any really) case to do with basic yang mills theory. We are considering a U(1) line bundle, L, ...
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95 views

Why are the fibers of principal G-bundles homeomorphic to G?

I'm trying to get a grip on the modern geometric formulation of gauge theory, in particular connections on principal G-bundles. However, I am stuck right after the definition already: Virtually all ...
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$G$-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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Does curvature zero mean the bundle is trivial?

Let $P\to M$ be some Bundle over $M$. I know that, if $P$ is a trivial bundle it must have curvature zero. Say I have the converse, my curvature is zero. Does this imply that the bundle ist trivial? ...
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106 views

Introduction a good text on principal bundle

Is there a good text that teaches principal bundle or frame bundle? I am looking for a textbook that might serve as an introduction to topology of principal bundles or frame bundles, specially the ...
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Equivalence between principal $ O(n) $-bundles and vector bundles

There's a well-known result (for example, Th. 14.2.7 in tom Dieck's book) that the category of principal $ \operatorname{GL}_n(\mathbb{R}) $-bundles and bundle maps is equivalent to the category of $ ...
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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 ...
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1answer
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Is the total space of a principal bundle parallelizable?

Given a smooth $G$-principal bundle $P \to M$, is $P$ in general parallelizable as a manifold? That is, is the tangent bundle $TP \to P$ trivial? In the case of Klein geometries, and more generally, ...
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Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
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Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
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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 ...
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Some doubts about the Atiyah groupoid

maybe this question is trivial, but I could not understand somethings about the Atiyah groupoid and algebroid. Let $\pi:P \twoheadrightarrow X$ be a principal $G$-bundle (where $G$ is a Lie group). ...
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How a principal bundle and the associated vector bundle determine each other

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...