# 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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### Is there a simple description of the total space of a principal S^1 bundle over a compact surface?

It is known that principal $S^1$-bundles over a compact surface $\Sigma_g$ are classified by their Chern classes in $H^2(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}$. When the Chern number is zero, the ...
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### Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?

Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
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### Connection on $U(1)$-bundle vs. 1-form

My original idea of a $\mathfrak{u}(1)$-valued connection $\omega$ is that it's simply a normal $1$-form. But in the middle of page 3 of https://www.arxiv.org/abs/math/0511710 he says "a ...
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### What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
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### Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
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### Spinor bundles and spin structures

I want to check my understanding of the relationship between spinor bundles and spin structures. So, as far as I understand, given a principal $\operatorname{SO}(n)$-bundle $\pi : P_{SO} \rightarrow M$...
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### Is the Hopf bundle $SU(2)/U(1)$ or the dual?

The Hopf bundle is defined as the quotient map of $S^3\subset \mathbb{C}^2$ by the action of $U(1)$ by multiplication $(z_1,z_2)\mapsto (e^{i\varphi}z_1,e^{i\varphi}z_2)$. It's known that it is the ...
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### Why $K_\nabla$ is the same as $K_{\omega_S}$?

Let $G\subset GL_n(\mathbb{R})$ be a Lie subgroup and let $M$ be a manifold. Consider $S$ a $G-$structure on $M$ and $\nabla$ a connection compatible with $S$. We know that if $E=E(P,V)$ is the vector ...
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### Ehresmann connections on general G-bundles?

There are two definitions of connections on bundles in terms of horizontal bundles: In the case of a principal $G$-bundle $P$, a connection is a subbundle $HP<TP$ such that $HP\oplus VP=TP$ and it ...
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### Why are Klein geometries flat?

A Klein geometry can be seen as a principal $H$-bundle of the form $G\to G/H$. It is known (cf. Sharpe and Chern's book) that Klein geometries are examples of flat Cartan geometries because their ...
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### The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
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### Coframe on coset space via connection one form

Suppose $G$ is a Lie group and $H$ is a Lie subgroup. Then, consider the principal $H$-bundle, $\pi:G\longrightarrow G/H$ where $G/H$ is coset manifold and $\pi$ is the canonical projection. Then, we ...
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### Why specifying values on a local section is enough to determine the local values of a tensorial form on a principal $G$-bundle?

Let $\pi: E \rightarrow B$ be a principal $G$-bundle. I have an initial context from which my question comes and I will explain how; I believe what I need is more general, but I may be wrong, and this ...
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### About equivariant vector valued forms on principal bundle

Let $\pi: M \to E$ be a $G$-equivariant vector bundle and let us adopt the notation $$C^{\infty}(M,E)^{G} = \{\tilde{s} \in C^{\infty}(M,E) | \ \forall g \in G \ \tilde{s} \cdot g = \tilde{s}\}$$ ...
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### Geometry of electrodynamics

In this question, I'd like to go over the physics - math dictionary occurring in the geometric structure (Principal bundle/spin bundles etc.) of Maxwell electrodynamics and the Dirac field. Consider ...
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### Vanishing of connection matrices for flat principal $G$-bundle

Background Recall that for a real vector bundle, there is a well known integrability theorem. Theorem. Suppose there is a vector bundle $E$ with fiber $\mathbb R^n$. If $A$ is a flat connection on $E$,...
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### Equating two definitions of principal fiber bundles

I am following these lectures on principal fiber bundles. Here, a principal fiber bundle is defined as a fiber bundle of which total space $P$ has a right free action of some Lie group and which is ...
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### Principal G-bundle induces a homeomorphism between orbit space and base

I am reading Chapter 14 of the algebraic topology book by Tammo tom Dieck. However, it is not going smooth since the very beginning. I will give the definitions first: Def (Principal G-bundle). Let $G$...
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### 3-connection on nontrivial 3-manifold

I'm studying Chern-Simons theory on topological nontrivial 3-manifold (I come from a physics background, so I'm new to some mathematical concepts). If the first homology group $H_1(M)$ is nontrivial ...
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### Does the monodromy action of a fiber bundle lie in the bundle's structure group?

Suppose we have a (topological) fiber bundle $p:E\to B$ with fiber $F$ and structure group $G$. Since $G$ acts on $F$ by homeomorphisms, it induces an action on the (integral) homology $H_*(F)$, i.e., ...
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### Are principal fibrations the same as group bundles?

I am reading Hatcher's algebraic topology, where he defines a fibration $F\to E\to B$ to be principal if up to choices of homotopy equivalences it can be written as $\Omega B'\to F'\to E'\to B'$ (with ...
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### Finding the group bundle associated to a $\mathbb{Z}[\pi_1]$ module

I am trying to understand twisted coefficients better, because although I technically know the definitions both using modules and group bundles, my intution of the subject doesn't really exist yet. To ...
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### Explicit formula for the principal connection 1-form induced by a Cartan connection

Let $P \subseteq G$ be a closed Lie subgroup. Suppose that a principal $P$-bundle $\mathcal{P} \to M$ is equipped with a Cartan connection $\omega: T\mathcal{P} \to \mathfrak{g}$. Then the extended ...
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### Exponentiate additive transition functions for $\mathbb{A}^1$-bundles.

Consider an smooth complex elliptic curve $E$ glued from two affine curves ($p\in\mathbb{C}\setminus0$) $$C_{(x,y)}: y^2=x^3+px\\C_{(s,t)}: t^2=ps^3+s$$ via the coordinate change $s=1/x,t=y/x^2$. It ...
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### Induced connection 1-form on orthogonal frame bundle by the Levi-Civita connection on tangent bundle

I recently posted a similar question but I have deleted that one and replaced it with this one, which is hopefully more focused. On page 317 of this paper by Pu-Young Kim and Joon-Sik Park they say ...
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### Understanding Associated Frame bundle terminology $u e_i = e_j^i X_j$

I am self-studying https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf and I am struggling with the definitions on page 37-38 Let us now see how the connection $\nabla$ manifests itself on ...
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### Valid interpretation of Higher order frame bundles and their jet groups?

I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid? NOTE: Always I ...
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### Why does the bar construction model the classifying space in both topology and AG?

For a topological group $G$, we can construct the classifying space $BG$ as the geometric realization of the nerve of $G$. I have seen a very similar assertion in the context of algebraic geometry: ...
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### A question related to frame bundle of a vector bundle

I am currently reading up on principal fiber bundles from a set of lecture notes on the subject, and I am trying to make sense of frame bundle of a vector bundle. Consider a vector bundle $p:E\to M$ ...
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### Doubts on definition of spin structure

In Friedrich's book "Dirac operators in Riemannian geometry" there is this definition for spin structure. Let $X$ be a CW-complex and $(Q, \pi, X, SO(n))$ a $SO(n)$-principal bundle on $X$. ...
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### Some other perspective of reduction of structure group

$(E,p,M)$ is a fiber bundle. Suppose there exists a local trivialization $LT_1$ of $E$ such that on the overlapping areas the transition can be realized by left action of a topological group $G$ (...
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