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Results tagged with connections
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user 3948
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].
6
votes
Accepted
Converging paths implies converging parallel transports along those paths?
The following is a precise version of the "convergence of equations implies convergence of solutions" claim.
Theorem. Let $\Omega\subseteq \mathbb{R}^n$ be open, and $Q\subseteq \mathbb{R}$. Let $A:\O …
- 39.1k
8
votes
Accepted
Symmetric Ricci Tensor
They are both correct. :-)
I gave a somewhat detailed write-up last year on my blog, but the gist of the argument is that if $\tau = 0$, then $d\tau = 0$. On the flip side, if $d\tau = 0$, locally yo …
- 39.1k
36
votes
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
After that, there is a characterisation of metric connections given by Schmidt, CMP 29 (1973) 55-59, which states that the linear torsion-free connection is metric if and only if the holonomy group is …
- 39.1k
6
votes
Rolling without slipping interpretation of torsion
Judging by your point (2), I think your picture is not entirely correct. The concept you are referring to is called a development. The idea is something like this (I "get" it for the torsion-free case …
- 39.1k
2
votes
Accepted
Proving the basic identity which implies the Chern-Weil theorem
(1) No, a connection is not a section of $\Omega^1(M,\mathrm{End}(E))$: a section would act tensorially and not satisfy the Leibniz rule. The connection is $\mathbb{C}$ linear and not $C^\infty(M,\mat …
- 39.1k