Thanks. I was thinking that always appears factors of the form $(f-f(p))(g-g(p))...$. Then if $f(p)=g(p)=0$, $v(f\cdot g)=\delta (v)(f,g)$, with the coproduct $\delta: {T^{\square}_p}^rM \longrightarrow {T^{\square}}^{r-1}M \otimes {T^{\square}}^{r-1}M $. It gives us the set of vectors $\{(u_i,w_i)\}$. I think this result it's not very useful.

Thanks! By the word fibration do you mean surjective submersion? In topology it means something different. I find this observation very beautiful. In Michor's book, there's a proof that all the fiber bundles (locally trivial surjective submersions) have complete connections. I think we can recover a complete connection from these complete locally connections. I was thinking if there is on another criteria similar than $\pi$ to be proper, but weaker.

Thanks Ben and Sebastian. I understand now more or less what you say about the image. If $\Gamma:E \longrightarrow J^1E$ is a section, then $\Gamma(\xi)=j^1_p\phi$ for some section $\phi$. The image is $d\phi_p(T_pM)\subset T_\xi E$, it's well-defined because depends of the germ of $\phi$ at $p$ not on the section. But I don't see why this collection forms a horizontal subspace.

Mmm, I see that I have to study more to completely understand it. The analyticity property of the action means that the action can be recovered by the Taylor coefficients? Is it necessary the action to be transtitive? You've mentioned that $(G,F)$ is a homogeneus space. Do you have any recommendation to read these properties about Jets and group actions?

Thanks. I'm not familiar with Jets enough to undesrtand it, but I think I catch the idea. Is it like calculate the "Taylor expansion" on $s$ of $g(x)\cdot s$ and recover $g(x)$?

Yes, my fault sorry. Thanks. I edited it