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Results tagged with connections
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user 20302
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].
11
votes
Why is it important that partial derivatives commute?
Here is another way of obtaining the Christoffel symbols with the symetry imposed by the torsion free condition
$$ \Gamma^i_{k\ell}=\Gamma^i_{\ell k}. $$
This goes back to Riemann's Habillitation. …
- 34.7k
5
votes
A geometric interpretation of the Levi-Civita connection?
The Levi-Civita connection is locally described by the Christoffel symbols. How does one obtain these in a natural fashion: write the Euler-Lagrange for the length functional. The extremals of thi …
- 34.7k
5
votes
Accepted
Terminology of "covariant derivative" and various "connections"
As I understand it, the "covariant"
part of this comes from the fact that
the T∗M component changes covariantly
under coordinate changes and not how
the E component changes. Is this
corre …
- 34.7k
7
votes
Accepted
The purpose of connections in differential geometry
The instantons are simply connections with minimal "energy". …