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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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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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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21 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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41 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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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 ...
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3answers
76 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
47 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
27 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
77 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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24 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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72 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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2answers
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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1answer
32 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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2answers
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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76 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 ...
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2answers
26 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 ...
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22 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 ...
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30 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
39 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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1answer
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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56 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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1answer
52 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 ...
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28 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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1answer
32 views

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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93 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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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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2answers
74 views

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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29 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 ...
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4answers
99 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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1answer
33 views

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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24 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 ...
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2answers
53 views

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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22 views

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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1answer
114 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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17 views

Reductions of structure groups and sections of coset bundles

I'm looking for a reference for the following proposition: Let $G$ be a Lie group and $H$ a (closed) Lie subgroup of $G$. Let $E \to B$ be a principal $G$-bundle. Then reductions of the ...
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105 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 $D$. ...
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57 views

Difference between types of connections [closed]

For my background, I am familiar with the basics of differential geometry, especially Riemannian geometry, and in some more advanced topics relevant to physics, especially general relativity. Lately ...
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90 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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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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1answer
56 views

Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
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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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1answer
74 views

Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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36 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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1answer
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Is a principal bundle automorphism locally given by a left action?

Let $G\hookrightarrow P \xrightarrow{\pi} M$ be a principal bundle, denote by $\cdot$ the right action of $G$ on $P$. Let $f:P\rightarrow P$ a bundle automorphism (i.e. $f$ is a diffeo, $f(p \cdot g) ...
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1answer
79 views

Is this operation meaningful or it is a mistake in the book?

I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is ...
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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 ...
5
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57 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 ...
4
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1answer
64 views

To Reconcile Two Different Descriptions of the Dual Bundle

$\newcommand{\mc}{\mathcal}$ Let $\pi:E\to M$ be a smooth vector bundle with typical fibre a $k$-dimensional vector space $\mc V$. There are (at least) two ways to construct the dual bundle of $E$. ...
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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. ...
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2answers
55 views

A Lie Group Homomorphism $f:G\to H$ Induces a Functor from Principal $G$-Bundles to $H$-Bundles

I am trying to understand Qiaochu Yuan's answer to this question. The first line of the answer reads: A Lie group homomorphism $f:G\to H$ induces a functor from the category of principal ...