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Results tagged with connections
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user 118688
Ehresmann connections; covariant derivatives; connections on vector bundles, principal bundles, ∞-bundles, submersions, bundle gerbes; holonomy and higher holonomy; parallel transport; torsion; curvature. See also the tags [principal-bundles], [vector-bundles], [gerbes], [curvature], [geodesics], [characteristic-classes], [torsion].
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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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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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Is there a notion of a connection for which the horizontal lift of a curve depends on its or...
Given a curve $\gamma$ on $M$ and fixing a point $x$ in the fibre of $\gamma(0)$, there exists a curve that starts at $x$. So, this lift has starting point as $x$. Suppose you choose $t\mapsto \gamma( …
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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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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. …
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3
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0
answers
468
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Applications of Ambrose-Singer theorem on holonomy
I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. …
2
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1
answer
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Advantages of Atiyah sequence version of connections on a principal bundle
Questions :
What are the other advantages of using Atiyah sequence to study connections? … Is there an account of Chern-Weil theory using Atiyah sequence definition of connections?
Is there an account for studying Characteristic classes of Principal bundles, using Atiyah sequences? …
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Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$
An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
…
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Accepted
determinant of curvature (notation issue)
Curvature $\Omega$ is a $\mathfrak{g}$ valued $2$-form on $P$ i.e., for each $p\in P$, we have $\Omega(p):T_pP\times T_pP\rightarrow \mathfrak{g}$.
As $\mathfrak{g}$ is $\mathfrak{gl}(r,\mathbb{C})$ …
1
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1
answer
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determinant of curvature (notation issue)
This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow …
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Characterisation of (integrable) connections on (trivial) principal bundle
The following result is from the book Differential Geometry - Bundles, Connections, Metrics and Curvature by Clifford Henry Taubes. … I am also interested in only "different" connections in the sense if two connections on $(P,M)$ are related by an isomorphism, in the sense of pullbacks, then I am calling these to be same. …
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References on principal G bundle and connections
Other books that helped me to learn more about principal bundles are
Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W. … Tu
Connections, Curvature, and Cohomology Volumes 1,2,3 by Werner Hildbert Greub, Stephen Halperin, James S. Vanstone, Ray Vanstone …
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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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2
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1
answer
879
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Characterisation of (integrable) connections on (trivial) principal bundle
Question : Is there a characterization of connections on $P(M,G)$; in the sense, a one-one correspondence between the set of connections on $P(M,G)$ and some "well-described set"? … A characterization for integrable connections on trivial principal bundle? …
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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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