I'm curious about possible equivalent formulations of the infinite jet bundle. It seems like all the jet bundles could be equivalently constructed by modding out by the equality of partial derivatives everywhere at once. You would lose the explicit bundle structure, but it would be functionally equivalent. Also, in the infinite jet bundle, it's hard for me to see why it doesn't contain all the structure of $C^\infty$. Here's the question more formally:
Let $M$ and $N$ be smooth manifolds. We define the following equivalence relations on $C^\infty(M,N)$:
- For $p\in M$ and $k\in\mathbb N$, set $uR_p^kv$ if the partial derivatives of $u$ and $v$ coincide up to order $k$ at $p$.
- For $p\in M$, set $uR_p^\infty v$ if all partial derivatives of $u$ and $v$ coincide at $p$.
- For $k\in\mathbb N$, set $uR^kv$ if the partial derivatives of $u$ and $v$ coincide everywhere up to order $k$.
- Set $uR^\infty v$ if all partial derivatives of $u$ and $v$ coincide everywhere.
The usual jet spaces are defined as $$J_p^k(M,N)=C^\infty(M,N)/R_p^k$$ whence the jet bundle is the fiber bundle $\pi^k:J^k(M,N)\to M$ such that $(\pi^k)^{-1}(p)=J_p^k(M,N)$.
The infinite jet space at $p$ is defined as the inverse limit of the $k$-jet spaces, $$J_p^\infty(M,N)=\lim_{\longleftarrow}J_p^k(M,N)$$ and the infinite jet bundle is the inverse limit of the $k$-jet bundles, $$J^\infty(M,N)=\lim_{\longleftarrow}J^k(M,N)$$
My question is, what is the relationship between the following sets of spaces:
- $J^k(M,N)$ and $C^\infty(M,N)/R^k$
- $J_p^\infty(M,N)$ and $C^\infty(M,N)/R_p^\infty$
- $J^\infty(M,N)$, $\displaystyle \lim_{\longleftarrow}C^\infty(M,N)/R^k$, $C^\infty(M,N)/R^\infty$, and just $C^\infty(M,N)$ itself.
I am afraid that the intuition you are following in this construction is not really correct. The jet space $J^k(M,N)$ is not a quotient of $C^\infty(M,N)$ in a reasonable sense. If you have $uR^k_pv$, then in particular $u(p)=v(p)$ (and otherwise it would not make sense to compare the partial derivatives at $p$). Thus $uR^kv$ if and only if $u=v$. It is just the fiber of $J^k(M,N)$ at a point $p\in M$ that is a quotient of $C^\infty (M,N)$ (and actually it would be more clear to only use germs of smooth functions at $p$ here). Describing the fiber in such a way is a short cut to avoid an involved algebraic description.
The object that $C^\infty(M,N)$ compares to in this context is the space of sections $\Gamma(J^k(M,N))$ or $\Gamma(J^\infty(M,N))$ of which it is a (rather small) subspace. The bundles $J^k(M,N)$ or $J^\infty(M,N)$ are in my opinion best thought of as extensions of the trivial bundle $M\times N$ (whose space of sections is $C^\infty(M,N)$). What one is doing in the construction (from the point of view of smooth functions) is to collect more information in a point than just the value. One e advantage of this is, that you can view differential operators as smooth functions defined on jet prolongations. To get the picture, it is probably instructive to look at $k=1$ in detail. Here the fiber over $p\in M$ can be viewed as pairs $(y,A)$, where $y\in N$ in any point and $A$ is a linear map from $T_pM$ to $T_yN$.This can be viewed as a formal version of the value and first derivative at $p$ of a smooth function, but there is no need to define it as a quotient of $C^\infty(M,N)$. In principle, one could define higher jet bundles in a similar way, but this would get pretty awkward since the standard tools of differential geometry are not well suited to deal with higher derivatives. The quotient $C^\infty(M,N)/R^k_p$ is just a highly efficient description for the finite dimensional space $J_p^k(M,N)$, but this only works in a point.