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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].
61
votes
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
Yes.
First, there's a very simple criterion for whether $\nabla$ is an orthogonal connection: look at the holonomy of $\nabla$ around closed
loops in the manifold, and ask whether they preserve a q …
- 27.5k
22
votes
Accepted
Rolling without slipping interpretation of torsion
The non-torsion-free connections are ones that
impart twists on little neighborhoods of curves. …
- 25.1k
12
votes
Accepted
Can a homotopy inverse of the map from a Lie group to loops on its classifying space be give...
However, to get a more immediate classifying space for connections that works
for all Lie groups, just make
a simplicial complex whose simplices are connections for a $G-bundles$ over the simplex. … Glue these simplices with $G$-bundles
and connections together, to make a model for $BG$. …
- 25.1k