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In Lectures on Integrability of Lie Brackets, Proposition 3.15, Fernandes and Crainic motivate what it means to say that two paths on an algebroid are homotopic. I spent quite some time trying to grasp the proof, but it still eludes me. I will write down some definitions as to make this question self-contained, but everything I write down can be looked up in their document.


  • For an algebroid $\pi : A \to M$ with bracket $[\cdot,\cdot]$ and anchor $\rho$, they define the notion of an $A$-path as a $C^1$-path $a : [0,1] \to A$ such that $\rho \circ a = \frac{d \gamma}{dt}$, where $\gamma = \pi \circ a : [0,1] \to M$ is the corresponding base-path.

  • A variation of $A$-paths is a family of $A$-paths $(a_\epsilon)_{\epsilon \in [0,1]}$ which is $C^2$ in $\epsilon$ and so that the end points of the base paths are fixed.

  • Lastly, if $\alpha(s,\cdot) \in \Gamma(A)$, $s \in [0,1]$ are time-dependent sections, their flow is the the unique maximal family of $A$-automorphisms $\{\phi^{t,s}_\alpha\}_{t,s}$ with $\phi^{t,s}_\alpha \phi^{s,u}_\alpha = \phi^{t,u}_\alpha$, $\phi^{t,t}_\alpha = \text{id}$ and \begin{align*} \frac{d}{dt} \Big \vert_{t = s} \left(\phi^{t,s}_\alpha\right)^\star (\beta) = [\alpha^s,\beta] \quad \forall \beta \in \Gamma(A) \end{align*} with $(\phi^{t,s}_\alpha)^\star (\beta)(x) = \phi^{s,t}_\alpha \beta(\phi^{t,s}_{\rho(\alpha)}(x))$, where $\phi^{t,s}_{\rho(\alpha)}$ is the usual time-dependent flow on $M$.

With these definitions in place, in Proposition 3.15 they are given a variation of $A$-paths $(a_\epsilon)$ and a time-dependent section $\xi_\epsilon$ so that $\xi_\epsilon(t,\gamma(t)) = a_\epsilon(t)$. In the proof, they define $$\eta (\epsilon,t,x) := \int_0^t \phi^{t,s}_{\xi_\epsilon} \frac{d \xi_{\epsilon}}{d \epsilon} (x,\phi^{s,t}_{\rho(\xi_\epsilon)} (x)) ds \in A_x.$$

They prove that the so-defined section $\eta(\epsilon,t,\cdot) \in \Gamma(A)$ has the property $$ \frac{d \eta}{d t } - \frac{d \xi_\epsilon}{d \epsilon} = [\eta,\xi].$$


Up to here, I think I have been able to follow the proof, but now I am lost: They define $X := \rho(\xi), Y := \rho(\eta)$ with the anchor of the algebroid, and say these fulfil a similar equation, presumably $$ \frac{dY}{dt} - \frac{dX}{d\epsilon} = [Y,X], $$ simply by applying the anchor map.
They then claim that because $X(\epsilon,t,\gamma_\epsilon(t)) = \rho( a_\epsilon(t)) = \frac{d \gamma_\epsilon}{d t}(t)$ (true as property of an $A$-path), it follows that $Y(\epsilon,t,\gamma_\epsilon(t)) = \frac{d \gamma_\epsilon}{d \epsilon}$. I have no idea how that follows and have tried a couple different unsuccessful approaches. It seems like this has something to do with $[Y,X](t,\epsilon,\gamma_\epsilon(t)) = 0$, which may well be true, but I cannot see a good reason. Apologies for this long, convoluted and very specific question, but I am kinda hoping that there is an absolute algebroid-pro running around who can instantly see the solution to my troubles. Thank you!

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