Let $X$ and $Y$ be two time-dependent vector fields on a manifold $M$. I found the following identity in a paper: $$\Phi^{X+(\Phi_{t, 0}^X)_*(Y_t)}_{t, s}=\Phi^X_{t, 0}\circ \Phi^{Y}_{t, s}\circ \Phi^{X}_{0, s}.$$ Does anyone know how to prove it? Is there a geometric interpretation for this identity?
Any reference woulbe also welcome.
Thanks.
Remark. A time dependent vectof field is a smooth map $X:\mathbb R\times M\longrightarrow TM$ such that $X(t, p)\in T_p M$ for every $(t, p)\in \mathbb R\times M$. Indeed, we could take as the domain of $X$ an open set of $\mathbb R\times M$.
Notice $X_t:M\longrightarrow TM$, $p\longmapsto X(t, p)$, is a vector field on $M$ for every $t\in\mathbb R$.
The time-dependent flow of $X$ is determined by the requirement $$s\longmapsto \Phi_{s, t}(p)$$ is the integral curve of $X$ starting at $p$ at time $s=t$, that is: $$\frac{d}{ds} \Phi^X_{s, t}(p)=X(s, \Phi_{s, t}(p))\quad \textrm{and}\quad \Phi^X_{t, t}(p)=p.$$ Then $\Phi_{s, t}: M\longrightarrow M$ is a diffeomorphism for every $(s, t)$ and the following identity holds $$\Phi_{s, r}^X\circ \Phi^X_{r, t}=\Phi_{s, t}\quad \textrm{and}\quad \Phi_{s, s}=\textrm{id}_M,$$ and consequently $(\Phi^X_{s, t})^{-1}=\Phi_{t, s}$.
To prove the existence of such a flow you must add more hypothesis because usual flows are defined only locally without further assumptions. But I guess this illustrates the concept.
Briefly, $\widehat{X}:=\frac{\partial}{\partial t}+ X$ is a vector field on $\mathbb R\times M$ so it has a flow and this flows satisfies $$\Phi^X_s(t, p)=(s+t, \Phi^X_{s+t, t}(p)).$$ This defines uniquely $\Phi^X_{s, t}$.