# 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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### Proof check: a riemannian 2-manifold is orientable if its unit tagent bundle is principal

For a two-dimensional smooth manifold $M$ with a Riemannian metric, we can consider the unit tangent bundle $S\to M$, which is a circle bundle over $M$. I am trying to show that if $S\to M$ is a "...
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### Connection one form and well-definedness of everything

I'm trying to learn differential geometry from Isham's book and I want to check if what I'm saying makes sense. Let $G \rightarrow P \rightarrow \mathcal{M}$ be a principle bundle. Then the connection ...
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### Constructing a connection $1$-form from local forms.

I am following Section $10.1.3$ of Geometry, Topology and Physics by Nakahara, and have ran in to an issue regarding local connection forms. Consider a principal $G$-bundle, $P(M,G)$, and an open ...
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### Explicit example of an Ehresmann connection on the trivial bundle $\mathbb{R}^4 \times S^1$

$\mathbb{R}^4 \times S^1$ is of course a principal $U(1)$-bundle and trivial. However, I cannot find an explicit example of the Ehresmann connection one-form on this bundle in terms of the coordinates ...
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### Principal $G$-bundle maps and sections of Associated Bundles

These days, I am making my way towards the classification result about homotopy classes of maps from a CW-complex $Y$ to $BG$ and isomorphism classes of principal $G$-bundles over $Y$, where $G$ is a ...
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### From when did people started assuming a principal bundle is locally trivial?

Question is as in the title: From when did people started assuming a principal bundle is locally trivial? I am asking this because, Dale Husemoller in his book "Fibre bundles" (1966) ...
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### Local trivializations and transition law of Atiyah Lie algebroid

Let $P\rightarrow M$ be a $G$-principal bundle and denote by $\mathrm{at}(P):=TP/G$ the Atiyah Lie algebroid over $M$. I want to understand how $\mathrm{at}(P)$ is a locally trivial Lie algebroid with ...
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### Properties of connection principal bundle

I am working on a proof of Cartan's formula $$DA(u,v) = dA(u,v) + [A(u),A(v)],$$ where $A$ is a connection one form on a principal bundle and $u,v$ are two vector fields. $D$ denotes exterior ...