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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.
5
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1
answer
399
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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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votes
3
answers
2k
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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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1
vote
0
answers
402
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Introducing connection on principal bundle as lifting of vector field and paths
Let $\pi:P\rightarrow M$ is a principal $G$ bundle. I want to introduce the notion of connection as a way to uniquely lift the structures on $M$ to structures on $P$, namely vector fields and paths.
…
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vote
0
answers
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references for holomorphic principal bundles (over complex manifolds)
principal bundles in differential geometry is a classical notion and there are so many references that discuss these notion (even in text books). But, when it comes to its version in complex geometry, …
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1
vote
3
answers
963
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Notion of Torsors
I am trying to read this paper by Lawrence Breen.
It starts with the definition of a torsor.
Let $G$ be a bundle of groups on a space $X$. The following definition of a principal space is standard, b …
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0
votes
Notion of Torsors
I have found the notion of group bundle in Alexandre Grothendieck's "A general theory of fibre spaces with structure sheaf".
This is for my own reference and may be useful for some one who reads tha …
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1
vote
Atiyah Sequence and Connections on a Principal Bundle
This is from another reference. So, adding as a different answer.
Appendix A "On principal bundles and Atiyah sequences" in the book Lie groupoids and Lie algebroids in differential geometry by Kiril …
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2
votes
0
answers
118
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Realization of a $\mathfrak{g}$-valued $2$-form as a curvature form
Consider the Lie group $S^1$. Recall that the associated Lie algebra is $\mathbb{R}$.
Let $M$ be a manifold. Consider the second de-Rham cohomology group $H^2(M,\mathbb{R})$.
Let $\Omega\in H^2(M,\ …
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Atiyah Sequence and Connections on a Principal Bundle
See the paper The Atiyah bundle and connections on a principal bundle by Indranil Biswas.
Let $p:E_G\rightarrow M$ be a principal $G$-bundle. Michael Atiyah in his paper uses a exact sequence of vect …
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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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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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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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2
votes
1
answer
442
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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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1
answer
278
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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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2
votes
0
answers
133
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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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