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Given an Ehresmann connection on a fibre bundle, is it possible to define a covariant derivative that measures the rate of change of a section of the fibre bundle as you move through the base manifold?

It seems that the usual covariant derivative on the tangent bundle $TM$ of the base manifold $M$ is a function $\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$ which takes a tangent vector field (the direction we are measuring the change in) and acts on another tangent vector field to produce a third tangent vector field. This of course can be generalised to tensor fields so you would have something like $\Gamma(TM) \times \Gamma(\bigotimes TM) \to \Gamma(\bigotimes TM)$. So you can measure the rate of change of a section of the tangent bundle or tensor bundle as you move along a curve in $M$.

For a general fibre bundle $B$ over a base manifold $M$, I think a covariant derivative would have to take a tangent vector from $TM$ and act on a section of the bundle $B$ to produce a section of the tangent bundle to the bundle, $TB$. So it would be a function $\Gamma(TM) \times \Gamma(B) \to \Gamma(TB)$. Now it seems that the desired derivative operation would take a curve with the tangent vector in $M$, lift that curve to $B$ so that it lies in the section, get the tangent to the lifted curve in $TB$, and finally return the projection of that lifted tangent vector onto the vertical bundle $VB$ (which is where the connection comes in).

Does this operation make sense?

And perhaps more importantly, is it meaningful? Does the vertical projection of a vector tangent to a section convey the intuitive meaning of "rate of change of the section"?

And finally, does it reduce to the usual covariant derivative if $B$ is the tangent bundle $TM$, and to the derivative of a scalar field if $B$ is just the trivial bundle $M\times \Bbb R$?

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  • $\begingroup$ what do you mean for $\bigotimes TM$? $\endgroup$ – janmarqz Oct 26 '15 at 21:34
  • $\begingroup$ By $\bigotimes TM$ I mean some tensor bundle on $M$ - a bundle where the fibres are tensor products of copies of the tangent space and cotangent space at each point in the manifold. $\endgroup$ – Matt Dickau Oct 26 '15 at 21:39

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