Covariant derivative vs. Ehresmann connection

I know about Ehresmann connections on fiber bundles and covariant derivatives as an (equivalent) way to define linear Ehresmann connections on vector bundles. My question is:

Is there any notion of covariant derivative equivalent to Ehresmann connection in the most general setting concerning fiber bundles?

When I say "the most general setting", I am emphasizing that the fiber bundle do not have any further structure than being just a fiber bundle (it may not be a vector bundle nor a principal bundle).

Diego

PS: I'm concerning the case when the fiber bundles are smooth. I don't worry about the non smooth case.

• Have you seen this question? Oct 6 '18 at 23:44

I think there is an analog of a covariant derivative associated to an Ehresmann connection, but it produces a more complicated object. Consider a fiber bundle $$p:E\to M$$ with fiber $$F$$, and a smooth section $$s:M\to E$$. Then you can take the tangen map $$Ts:TM\to TF$$ and use the Ehresmann connection to projctet to the vertical subbundle $$VF=\ker(Tp)\subset TF$$. So for each $$x\in M$$, this produces a linear map $$T_xM\to V_{s(x)}F$$ that can be interpreted as a smooth section of $$T^*M\otimes s^*VF$$, which is a good analog of the standard covariant derivative. Alternatively, you can insert a vector field $$\xi$$ on $$M$$ to obtain a section $$\nabla_\xi s$$ of $$s^*VF$$.
This exactly reproduces the standard concept in the case that $$F$$ is a vector bundle, since in that case $$VF$$ can be naturally identified with $$p^*F$$ and thus $$s^*VF$$ is canonically identified with $$F$$ for any section $$s$$.