# 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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### Principal bundles over affine schemes

I came up with the following question. Let us suppose that we have an affine scheme $X=\text{Spec}(A)$. It is well known that vector bundles on $X$ are equivalent to $A$-modules. In particular, if ...
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### Existence of smooth sections in principal bundles.

Consider a smooth principal bundle $(P, M, \pi, G)$. For every open $U \subset M$ (except $M$ itself) in the base space $M$ is there a local section? I.e, a smooth map $s: U \rightarrow P$ that ...
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### Equivalence between fiber bundles with fiber F, and principal Aut(F)-bundles

First, let's fix some definitions. Definition 1: Let $G$ be a topological group. A principal G-bundle is a triple $\xi = (E,\pi,B)$, where $E$ is a right $G$-space and the action of $G$ on $E$ is free ...
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### On characterisation of smooth $G$ equivariant morphisms between Product manifolds with $G$ action

In particular I am interested in the following! Let $M$ be a smooth manifold and $G$ be a Lie Group. Let $\rho: (M \times G)\times G \rightarrow M \times G$ be the smooth action of $G$ on $M \times G$ ...
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### Tangent space of a Principal bundle

Suppose we have a Principal bundle with a Lie group G in fiber. It is known that through the trivializations it can locally be expressed as a product of the base space M and the group in the fiber. ...
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### Isomorphism between basic forms on principal bundle and forms on the associated vector bundle?

$P$ is a principal $G$-bundle, $ρ:G→GL(V)$ is a representation, and $E$ is the associated vector bundle of $P$, so $E=(P×V)/G$ with the right action $(p,v)⋅g=(p⋅g,ρ(g^{−1})v)$. I have read that ...
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### Modifying trivializations in principal bundles

I am trying to figure out when modifying the trivializing neighborhoods of a principal (or just locally trivial) bundle alters the global bundle structure. As a practical example in which I ...
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### When is a functor the associated bundle construction?

Let $G$ be a group and $X$ be a space. I am principally interested in these objects in the category of schemes over some base $S$, but an answer in another geometric category would be welcome. ...
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### Under which conditions every 2-form is a curvature form?

Let $M$ be a smooth manifold, $G$ a Lie group and $P\rightarrow M$ a smooth principal $G$-bundle. Let $\Omega^1_{eq} (P;\mathfrak{g})$ denote the space of $G$-equivariant $\mathfrak{g}$-valued 1-forms ...
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### Transition functions of $S^3$ as a principal bundle over $\mathbb{C}P^1$.

Consider he action of $U(1)$ on $S^3$ by $(z_1,z_2)\cdot e^{i\phi}\mapsto(z_1e^{i\phi},z_2e^{i\phi})$. This action is clearly free and proper. Therefore we can consider $S^3$ as a principal bundle ...
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### Frame bundle of associative vector bundle returns the origional prinicipal bundle.

Background and definitions: For a vector bundle $E$ let $F(E)$ denote its frame bundle. For a principal bundle $P$, and representation $(\rho,GL(n,\mathbb{R}))$ let $E(P,\mathbb R^n)$ denote the ...
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### Immersed principal subbundles

In Kobayashi and Nomizu's Foundations of Differential Geometry vol. 1 the authors define immersed principal subbundles (actually, they call them imbeddings) of a principal $G$-bundle $\pi: P \to M$ as ...
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### Associated vector bundles: Given a vector bundle $E$ show that $F(E)\times \mathbb{R}^n/GL(n,\mathbb{R})\cong E$

I am trying to understand the map between vector bundles and principal fibre bundles and would like to work out the following example explicitly. Moreover in this example I am only currently looking ...
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### When is a connection on the adjoint bundle induced by a principal connection?

Let $P\rightarrow M$ be a principal $G$-bundle with a connection 1-form $\omega$. In a local trivialisation $\tau_U \colon U\rightarrow P_U$ ($U \subset M$) we can pull the connection back to the ...
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### Isomorphism between space of differential forms and space of basic forms on pricipal $S^1$-bundle and independence of cohomology class

Let $M$ be be a principal $S^1$-bundle, i.e. a smooth manifold with a smooth $S^1$-action (multiplication to the right) that has no fixed points such that the orbits through every point $p \in M$, ...
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### Differential operators of a principle G-bundle

I understand for a pair of smooth vector bundles $E$ and $F$ over a smooth manifold $M$ it makes sense to talk about the differential operators between the section spaces of $E$ and of $F$. The ...
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### Thin homotopy invariance of parallel transport

I have seen many claims that parallel transport in thin homotopy invariant, but I cannot seem to prove it. My conventions are: Let $\pi\colon E\to B$ be a principle $G$ bundle with connection $\omega$...
### Are classifying spaces of $GL$, $SL$, $O$ the same?
In the theory of principal bundle classification, we have the Milnor's construction $EG$ for a topological group $G$ and a classifying space $BG$. We know the classifying space for $GL_n(\mathbb{C})$ ...