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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 $G$.

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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, ...
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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 ...
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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 ...
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Covariant derivative: QFT vs. Math

In class, we have seen that the covariant derivative of some form $R$ can be written as: $$DR = dR + [A, R] = dR + A\wedge R - R\wedge A \tag1$$ Here, $d$ represents the external derivative over ...
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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\...
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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 ...
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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 ...
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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 ...
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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 $E$ ...
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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 \...
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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 ...
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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., ...
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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. ...
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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)= (\...
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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}...
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G-principal bundle and homotopy retract

Suppose that $f:X\rightarrow Y$ a continuous map between (connected) CW-complexes such that there exists a continuous map $g:Y\rightarrow X$ with the property that $g\circ f$ is homotopy equivalent to ...
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Showing the Hopf fibration has no global sections

Let's consider a principle $U(1)$-bundle over $S^2$ with the transition function $g_{\infty 0} = z/|z|$ (it is known as the Hopf fibration). There is a simple topological argument showing that this ...
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Showing simple transivity of action on fibre

Let $(P,X,\pi, G)$ be a principal $G$-bundle (definition below). I want to show (a) $\pi(u\cdot g)= \pi (u)$ (b) If $\pi(u)=\pi(v)$ then, there exists unique $g\in G$ such that $u=v\cdot g$. (a) ...
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Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
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Transitivity of the action in definining principal bundles

In the wikipedia article https://en.wikipedia.org/wiki/Principal_bundle the definition of a principal $G-$bundle $\pi:P\rightarrow X$ demands that the action of $G$ on $P$ to be free and transitive. ...
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Correspondence between flat connections and fundamental group representations

Let $M$ be a manifold. Two stackexchange posts state that there is a correspondence $$ \{ (P,A): P \text{ a $G$-bundle}, A \text{ flat connection} \} \leftrightarrow \{ \text{morphisms } f:\pi_1(M) \...
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Universal Bundle — Understand the basic definition

I read two versions of discussions on universal bundles. I could not really see how the two definitions are really the same. From Wiki. The universal bundle in the theory of fiber bundles with ...
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Two notions of classifying space

Why is the categorical classifying space for a group G, i.e., geometric realization of the nerve of G(as a category of One object), the same as the topological classifying space for principle G ...
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Line bundles associated to principal circle bundles

Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a ...
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Transitive action of $H^2(M;\Bbb Z)$ on $Spin^c$ structures over $M$

I’ve a problem understanding why the action of the second cohomology group (integer coefficients) of an oriented smooth manifold $M$ is free and transitive on the set of $Spin^c$. I’m following these ...
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If $H\leq G$, then $BH \to BG$ is a fiber bundle with fiber $G/H$

Suppose $G$ is a topological group and $H\leq G$ is a closed subgroup. The inclusion $H\to G$ induces a map on classifying spaces $BH\to BG$. I've seen in some sources that $BH\to BG$ is actually a ...
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Classification of principal bundles and of regular coverings

Let $G$ be a discrete group and let $X$ be a good space with fundamental group $\pi_1$. We know the following things: The connected principal $G$-bundles over $X$ are exactly the regular coverings of ...
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From sheaf torsors to geometric bundles on schemes

$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\Sym}{Sym}$ $\newcommand{\func}{\mathcal{O}}$ $\newcommand{\M}{\mathcal{M}}$ It is well known that the notions of locally free $\func_X$-...
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How to arrive at frame bundle objects from chart calculation on the base manifold

In this lecture by Fredric Schuller it is said that the wave-function is not a wave-function. He attempts to find an appropriate coordinate independent derivative using "chart calculations" on the ...
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How to recover the covariant derivative from the pull back from that on the principal bundle

I am watching these lecture series by Fredric Schuller. Covariant derivatives - Lec 25 - Frederic Schuller @minute 01:10:11 When we arrive at the covariant derivative from the principal bundle $P$ ...
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Exterior Covariant Derivative - Proof of Structure Equation for general r-form

Let $(P,M,\pi,G)$ be a principal bundle and $\omega \in \Omega^{1}(P,\mathfrak{g})$ a principal connection. Given a representation $\rho : G \to GL(V)$ and an equivariant form $\eta \in \Omega^{r}(P,V)...
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Mathematical challenge for unification of gravity and electromagnetism in classical theory?

I am trying to better understand the mathematical foundations of a possible reconciliation between quantum field theory and gravity in general relativity. However, before the application of the ...
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Question regarding global sections of principal bundles over a smooth oriented surface

I was reading a proof of the following theorem by A. Ramanathan : Let $X$ be a smooth connected oriented surface and $G$ be a connected topological group. Then there is a bijection between the set of ...
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Pullback of local sections over the total space

Let $\pi\colon P\to X$ be some $G$-principal bundle; $\{U_\alpha\}$ a cover of $X$; and $\{s_\alpha\colon U_\alpha\to\pi^{-1}(U_\alpha)\}$ a collection of local sections. Claim: The pullback $\pi^*P$ ...
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Spherical coordinates on the 2-sphere and k-forms

Consider the 2-sphere $S^2$inside $\mathbb R^3$, and let \begin{align} S:(0,\infty)\times(0,\pi)\times(0,2\pi)&\to\mathbb R^3\setminus\{(0,0,0)\} \\ (r,\phi,\theta)&\mapsto(r\sin\phi\cos\theta,...
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Space of principal connections is affine modelled on $\Lambda^1(M;\mathfrak{g})$?

I'm working within the jet-formulation espoused by Saunders in "The Geometry of Jet Bundles" and am struggling to prove the stated result. I would like to stay in this context and understand the ...
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Proof of a lemma concerning pull-backs of bundles

One has the following definitions: Let $f: M \rightarrow N$ be a smooth map between manifolds and let $\pi: P \rightarrow N$ be a $G$-principal bundle over $N$. The $pull-back$ of $P$ by $f$ is ...
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How to interpret $\nabla_i = \frac{\partial}{\partial x_i} +A_i$ in terms of a connection $A$ on a principal bundle.

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 ...
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1answer
29 views

Reading request: induced representations & fiber bundles

To my understanding, the theory of induced representations of a group $G$ can be formulated in terms of vector bundles associated with the group considered as a principal bundle ${G \to G/H }$. I'm ...
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Recovering a principal connection from its monodromy

Given a principal bundle $P \to M$ with structure group $G$ ($M$ and $G$ are connected), it is well known that one can recover the data of $P$ and a flat connection $\Gamma \in \Omega^{1}(P, \mathfrak{...
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Is $G$ action on a principal bundle a smooth submersion?

Let $\pi : P \rightarrow M$ be a principal $G$ bundle. Suppose $R_{g}: P \rightarrow P$ is a smooth map induced by the smooth right $G$ action on $P$. Is $ R_{g} $ a smooth submersion for all $g \in ...
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Lift of principal bundle over a compact Lie group

Suppose $\alpha:M\to N$ is a homomorphism of compact Lie groups and $\eta=(E,B,N)$ is an $N-$principal bundle over a compact manifold. Give a condition of $\alpha$ such that $\eta$ can lift as an $M-$...
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Is a principal bundle of a principal bundle still principal?

Let $\left(P_1,\pi_1,M,G_1\right)$ and $\left(P_2,\pi_2,P_1,G_2\right)$ be two principal bundles, where $M$, $P_1$ and $P_2$ are differential manifolds, and $G_1$ and $G_2$ are Lie groups. With these ...
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Is a certain subset of a group acting on a Hausdorff space closed?

Let $G$ be a topological group acting continuously on a Hausdorff space $X$. For $K\subset X $ compact, define \begin{equation} G_K=\{ g\in G : g K\cap K \neq \varnothing \}.\end{equation} Is $G_K$ ...
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Question concerning the relation between the reduction of a $G$-principal bundle to some subgroup $H$ of $G$ and Čech cocycles.

Let $G$ be a Lie group and $M$ be a smooth manifold. I have to show that a $G$-principal bundle admits a reduction to some subgroup $H$ of $G$ if and only if there exists a Čech cocycle $\{(U_{\...
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How does one introduce characteristic classes

How do you introduce or how are you introduced to characteristic classes. I am assuming the student is comfortable with principal bundles and connections on principal bundles.
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References: Equivalence between local systems and vector bundles (with flat connections)

I am working on a topic related to the Riemann-Hilbert Correspondence, and I was wondering if there is a reference that explicitly goes over the following equivalence of categories (when $M$ is a ...
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References for endomorphism bundle and adjoint bundle

I am trying to understand what are endomorphism bundle(of a vector bundle) and adjoint bundle(of a principal bundle) but could not find any references on google. Searching Adjoint bundle gives https:...
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Matrix representation of connections and curvatures on a principal(vector) bundle

Let $E\rightarrow B$ be a vector bundle and $\nabla$ be a connection on this vector bundle. I read some where that this connection and it’s curvature form can be represented by a matrix. I do not ...
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Associated principal bundle for a given vector bundle

The book I am referring is Foundations of Differential geometry by Kobayashi and Nomizu. In second volume, chapter on Characteristic classes, they write the following. Let $E$ be a complex vector ...