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Results tagged with connections
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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].
3
votes
An identity for the higher form Levi-Civita connection
Both $m∘∇$ and $d$ are natural operations from $k$-forms to $(k+1)$-forms.
By Palais's theorem, all such operations are proportional to the de Rham differential.
That is, $m∘∇=λd$ for some $λ$.
By exa …
- 37.8k
9
votes
Obstructions to the existence of a flat connection on a vector bundle
A $d$-dimensional flat real vector bundle $E→M$ is classified by a map $\def\B{{\sf B}}\def\GL{{\rm GL}}M→\B\GL(d)_δ$, where $\GL(d)_δ$ is the orthogonal group equipped with the discrete topology.
Arb …
- 37.8k
1
vote
Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?
For example, if $G$ is an ordinary Lie group, then connections on the trivial $G$-bundle are given by Lie algebra-valued differential 1-forms, which do possess the extension property from spheres to disks …
- 37.8k
18
votes
Accepted
A non-Abelian de Rham complex?
What is being described in the main post is simply the de Rham (crossed) complex valued in a Lie group (not necessarily commutative).
See, for example, Section 6.2 in Anders Kock's Synthetic Geometry …
- 37.8k
6
votes
Accepted
What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bun...
Contrary to what is claimed in the comments, I would argue
that the definition given in nLab's Idea section is rigorous
enough to be an actual definition in a research-level paper,
possibly with an ad …
- 37.8k
5
votes
1d TQFT minus connection =?
Furthermore, there is an equivalence between 1-dimensional smooth TFTs
over X and vector bundles with connections over X. …
- 37.8k
4
votes
Accepted
Flat connections, curvature and holonomy
The Stokes theorem must be modified first to deal with the nonabelian case.
See http://arxiv.org/abs/0802.0663,
Section 3.2, Theorem 3.4 and the displayed formula on top of page 48
for an appropriate …
- 37.8k