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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.
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0
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819
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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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References on principal G bundle and connections
I realised one can not (should not) escape from reading Kobayashi and Nomizu’s book.
Other books that helped me to learn more about principal bundles are
Differential Geometry: Connections, Curvatu …
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Characterisation of (integrable) connections on (trivial) principal bundle
This is not an answer. This is in response to Mike Miller's comment.
Let $M$ be a manifold, $\tilde{M}$ to be its associated universal cover (a simply connected covering space over $M$). I do not und …
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2
votes
1
answer
879
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Characterisation of (integrable) connections on (trivial) principal bundle
Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.
Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq …
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What is the geometric description of the set of isomorphism class of $G$-torsors over a site...
This is not a complete answer, too long for a comment.
If we start with an arbitrary site $\mathcal{C}$ and if we want to define the notion of a $G$-torsor over $\mathcal{C}$, then $G$ is not expecte …
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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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4
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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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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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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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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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2
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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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2
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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
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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3
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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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0
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393
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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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