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Results tagged with principal-bundles
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user 118688
A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.
9
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6
answers
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Motivation for construction of associated fiber bundle from a principal bundle
Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fiber bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G …
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votes
2
answers
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What does reduction of structure group of principal bundle say?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle.
The notion of reduction of structure group is standard but I will rec …
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3
answers
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Atiyah Sequence and Connections on a Principal Bundle
Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text …
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7
votes
1
answer
782
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Classification of Principal $G$ bundles and vector bundles in smooth sense
Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that
there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a dece …
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5
votes
1
answer
543
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Holonomy map on a connected manifold determines the connection and the bundle
I am reading Parallel transport on principal bundles over stacks. I quote from their paper :
Let $G$ be a Lie group and $M$ a $C^{\infty}$ manifold. Recall that a
choice of a connection $1$-for …
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5
votes
1
answer
399
views
In what sense bibundles are called as generalized morphisms
Definition : Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with two maps $a_L:P\rightarrow \mathcal{G}_0,a_R:P\righta …
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4
votes
0
answers
393
views
Chern-Weil theory and Weil homomorphism of principal bundle
In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to defin …
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4
votes
1
answer
278
views
Chern -Weil map for topological principal G bundles
Let $G$ be a Lie group.
In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :
The notion of a topological principal $ …
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4
votes
1
answer
234
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Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\right...
Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
See that the morphism of Li …
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3
votes
1
answer
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How does one introduce characteristic classes [closed]
How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not as …
3
votes
1
answer
681
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Principal bundles and fibre bundles
Let $\pi_P:P\rightarrow M$ a principal $G$ (right action) bundle. Let $F$ be a manifold with a left action of $G$. Then we have the notion of associated fibre bundle over $M$ whose fibre is $F$. I do …
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3
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4
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Alternative (easier) Proof of Ambrose Singer Holonomy theorem
Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ …
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4
answers
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References on principal G bundle and connections
I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, a …
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2
votes
1
answer
442
views
Advantages of Atiyah sequence version of connections on a principal bundle
I am reading Lie Groupoids and Lie Algebroids in Differential Geometry
by Kirill Mackenzie.
In appendix (page $291$), before discussing about Atiyah sequence associated to a Principal bundle, the aut …
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2
votes
0
answers
133
views
Associated bundle construction and classifying space
Let $\theta:G\rightarrow H$ be a morphism of Lie groups.
Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\ri …
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