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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].
18
votes
What is the Levi-Civita connection trying to describe?
Without the torsion-free assumption, there are many possible connections, and any properties derived from them will not be invariants of the metric. … People have studied connections that are not torsion-free, but so far the theory developed in that direction has not paid off nearly as much as Riemannian geometry has. …
- 27.5k
9
votes
Alternative (easier) Proof of Ambrose Singer Holonomy theorem
I always find it easier to work with the vector bundle induced by a linear representation of the structure group. I believe this theorem is a consequence of the following loop formula (a terse proof c …
- 27.5k
10
votes
Why is it important that partial derivatives commute?
If you don't assume the connection is torsion-free, then there are an infinite number of connections that are compatible with the metric (instead of exactly one), so the link between the geometric properties …
- 27.5k
12
votes
Accepted
Tensor contraction and Covariant Derivative
I like the question. Below is a somewhat sketchy version of how I see this.
I think the importance of tensors and contraction of tensors originates from trying to do basic differential geometry or ve …
- 27.5k
18
votes
pull-back connection
Here's my summary of the situation:
1) First, observe that the space of local sections of the pullback bundle is generated by the space of sections of the original bundle composed with the map $F$. ( …
- 27.5k