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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.