I have been stuying some problems in vector bundles and the concept of the covariant derivative of a vector bundle map comes up. However I cannot find a reference for this and hence would like to see if anyone can enlighten me.
That is consider for example $\eta,\xi$ to be smooth vector bundles over $\mathbb{R}\times S^1$ and $\Psi: \xi\rightarrow \eta$ to be a smooth vector bundle map. What is meant by $\nabla_{s}\Psi_s$? What is this covariant derivative and what properties does it have ? For example the relation $\partial_s (\Psi_s X(s))= (\nabla_s \Psi_s)(X(s))+\Psi_s(\nabla_s X(s))$, where $X$ is a section , is used.
I guess one idea I had was to consider $\Psi$ as an element of the endomorphism bundle $End(\xi,\eta)$ and then perhaps there is an induced connection on $End(\xi,\eta)$ from the ones in $\eta$ and $\xi$ such that $\nabla^{\eta }_ s \Psi X= (\nabla^{End(\xi,\eta)}_s \Psi)X+ \Psi(\nabla_{s}^{\xi}X)$. Is this going to be the case ? Perhaps we could define the connection by that relation exactly and hence get the desired reuslt straight way.
Any help is appreciated, thanks in advance.
Question: "I guess one idea I had was to consider Ψ as an element of the endomorphism bundle End(ξ,η) and then perhaps there is an induced connection on End(ξ,η) from the ones in η and ξ such that ∇ηsΨX=(∇End(ξ,η)sΨ)X+Ψ(∇ξsX). Is this going to be the case ? Any help is appreciated, thanks in advance."
Answer: If $E,F$ are finite rank vector bundles with connections $\nabla_E, \nabla_F$, there is a "canonical" connection $\nabla$ on $Hom(E,F) \cong E^*\otimes F$ - the tensor product of $\nabla_E$ and $\nabla_F$. Given any $\phi \in Hom(E,F)$ and any vector field $x$ you have thus a definition of $\nabla(x)(\phi)$. It has the property that for any section $s$ it follows
$$\nabla(x)(s\phi)=s\nabla(x)(\phi)+x(s)\phi.$$
You may define
$$\nabla(x)(\phi):= \nabla_F(x) \circ \phi - \phi \circ \nabla_E.$$