On the smooth structure of the spaces of $k$-jets

Question

I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.

1. the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, for any $f\in C^{\infty}(M,N)$;
2. the map $(\alpha,\beta):J^k(M,N)\to M\times N$, defined by $(j^k_x f)\mapsto (x,f(x))$, is a smooth summersion, for any smooth manifolds $M$ and $N$;
3. the composition map $\gamma:J^k(N,O)\times_{N} J^k(M,N)\to J^k(M,O)$, defined by $(j^k_{f(x)} g,j^k_x f)\mapsto j^k_x(g\circ f)$, is smooth, for any smooth manifolds $M$,$N$ and $O$, (here $J^k(M,N)\times_{N} J^k(M,N)$ is the fiber product of $\beta:J^k(M,N)\to N$ and $\alpha:J^k(N,O)\to N$);
4. the map $(\alpha,\beta)^{-1}(U,V)\to J^k(U,V)$ defined by $j^k_x f\mapsto j^k_x(f|_{U\cap f^{-1}(V)})$ is a smooth isomorphism, for any open subsets $U\subset M$, $V\subset N$;
5. for any open subsets $U\subset \mathbb{R}^m$ and $V\subset \mathbb{R}^n$, the map $J^k (U,V)\to U\times V\times \bigoplus_{i=1}^k{L^i_{sym}(m,n)}$, given by $j^k_x f\mapsto (x,f(x),Df(x),\ldots,(D^kf)(x))$, is a smooth isomorphism, (here $L^i_{sym}(m,n)$ is the vector space of the $\mathbb{R}^n$-valued symmetric $k$-multinear maps on $\mathbb{R}^m$).

Probably it is not sufficient, or redundant, but, in such a case, I would know if there is in the literature such a kind of characterization.

My question is: Once prescribed the usual smooth structure on the $J^k(U,V)$, for arbitrary open sets in euclidean spaces $U$ and $V$ (as in point 5), what kind of conditions are sufficient to uniquely determine the usual smooth structure on $J^k(M,N)$ for all other smooth manifolds $M$ and $N$?

Answer 1

Let $(U,u)$ is a chart for $M$, and $(V,v)$ be a chart for $N$. $u: U\to u(U)\mathbb R^n$ is diffeomorphism. $u(U)$ and $v(V)$ are open subset of $\mathbb R^n$ and $\mathbb R^m$. Then we can identify $$J^k(u(U),v(V))= u(U)\times v(V)\times \Pi_{j=1}^k L^j_{sym}(\mathbb R^n, \mathbb R^m)$$ People give manifold structure on $J^K(M,N)$ by chart $(J^k(U,V), J^k(u^{-1}, v))$. Main aim is to define map $J^k(u^{-1}, v)$. $$J^k(u^{-1}, v): J^k(U,V)\to J^k(u(U), v(V))\text{ is defined as following:}$$

Firstly for $u:U\to u(U)$ define $J^k(u,V):J^k(U,V)\to J^k(u(U),V)$ by $ J^k(u,V)j^kf_x= j^k(fog)_{g^{-1}(x)}$. This is a well defined map and $J^k(u,V)^{-1}= J^k(u^{-1},V)$

Now same way for $v$ define map $J^k(U,v): J^k(U,V)\to J^k(U, v(V)$. Take $$J^k(u^{-1}, v):= J^k(u^{-1},v)oJ^k(u(U),v)$$ This will be bijective and satisfy coordinate transformation condition:

For details please see: First Chapter 1.1 to 1.8 of Manifolds of differential mapping: P.W. Michor.

Answer 2

As I understand your question, the answer is: for any open set $U'\subset M, 'V\subset N$ such that $U,V$ are diffeomorphic to $U',V'$ you can identify $J^k(U,V)$ with subset of $J^k(M,N)$. So, it is enough that all such identivication are diffeomorphisms.