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 $N$?


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.


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.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.