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Results tagged with connections
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user 1648
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].
111
votes
6
answers
16k
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When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla_g$, the Levi-Civita connection, that is compatible with the metric.
I was wondering if one can …
- 63.9k
4
votes
1
answer
614
views
"Nash Style" Embedding Theorem for Connections
The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian manifo …
- 8,098
3
votes
1
answer
938
views
Holonomy Groups and the Hopf Fibration
I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a con …
- 33.3k