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Results tagged with connections
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user 14515
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].
15
votes
Accepted
Is there a way to define a Lie derivative of a connection?
It is because derivative is more or less the same as difference and difference of two connections is a (1,2)-tensor field. …
- 4,787
23
votes
Manifolds admitting flat connections
But of course there are manifolds that do not admit flat affine connections, the simples example is the sphere of any dimension $n>1$. …
- 4,787
6
votes
Accepted
Existence of connections making a bundle endomorphism parallel
My answer is almost contained in the answers/comments of Mariano Suárez-Alvarez, Ben McKay and Robert Bryant; I summarize the answers and give a reference.
There exists a torsion-free connec …
- 4,787
1
vote
Tensor contraction and Covariant Derivative
It is a natural requirement and is more-ore less equivalent to the natural analog of the leibniz rule. Let us consider the following example:
$$
d g(v,u) = \nabla g(u, v)+ g(\nabla u, v)+ g(u, \nabla …
- 4,787