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Let $\pi:E\rightarrow M$ be a vector bundle, and let $U\subseteq M$ be a trivialization domain for $E$. Assume a linear connection is given on $E$ with local connection form(s) $\omega=(\omega^a_{\ b} …

A particular case I am interested in is the following: The affine bundle $\mathcal C(P)\rightarrow M$ of principal connections on $P$ is a $W^{1,1}P$-associated bundle. …

It is a well known fact that then there are naturally induced linear connections on "related" vector bundles, such as $E^\ast,\ \bigotimes^kE$, etc. … Question 2: How is this process related to the usual way of inducing connections on vector bundles associated to principal bundles? …

It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop. …

This question is inspired by this physics stack exchange post, which is recent and has not received an answer yet, nontheless I feel that there is a better way to ask this question here with a larger …

If an appropriate set of connections are introduced then I am looking to rewrite this as $$ j^\infty\Xi\ \lrcorner\ \theta=\sum_{k=0}^{r-1}\hat Q^{ii_1...i_k}_\rho\nabla_{i_1...i_k}\Xi^\rho\mathrm d^{m … Some specific points of uncertainty are what connections are precisely required to carry out this procedure. …

Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following: If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, t …

I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann connections … on the (orthonormal) frame bundle in a rigorous manner, rather than Koszul connections on the tangent bundle, and develops calculus directly on the frame bundle, rather than on spacetime itself. …

. $$ Based on what I have read about Cartan connections, one can describe a Cartan connection modelled on $G/H$ by having a $G$-fiber bundle $(E,\pi,M,G/H,G)$ with typical fiber $G/H$, an Ehresmann $G$ …