Consider the following claim:
Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth
manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p_*:M^J \to
N^J$ is a submersion of Frechet manifolds, where $X^J$ denotes the usual manifold of smooth paths in $X$.
The intuition is that given a lift of a path $\gamma$ through $p$, one can find
a neighbourhood $U$ of the image of $\gamma$ and a neighbourhood $V$ of the image of
the lift such that for every path in $U$ one can smoothly choose a lift to $V$.
Here I suppose we need a submersion of Frechet manifolds to be a map that
admits local sections through every point in the domain, if that is
the 'correct'
notion of submersion in that setting (certainly not the 'surjective on tangent
spaces' version).
I think the proof would use the characterisation of submersions as maps
which look locally (on both the domain and codomain) like projections
$U \times \mathbb{R}^n \to U$, and the existence of good open covers with smooth
contractions.
I think I'm able to prove that there are continuous sections through every point
in the domain, thinking of everything as a topological space, and using the
compact-open topology on the mapping spaces. But I don't know off the top
of my head that the compact-open topology on the space of smooth paths
is the same as the topology inherited from the Frechet manifold structure.
(My guess is that it is.)
My question: is the claim true?
As Andrew Stacey points out in the comments, the mapping space is not a manifold for non-compact intervals. However, I think I really only need maps which have all derivatives uniformly bounded (but a different bound for each derivative!). Since the topology on the mapping space for compact intervals uses uniform convergence, I'm betting that this set has the structure of a Frechet manifold.
Question 2: am I right?
Question 1': if so, is the claim true for this (putative) map of Frechet manifolds?
Best Answer
This is answered affirmatively in Yet More Smooth Mapping Spaces and Their Smoothly Local Properties. Specifically:
Theorem 1.1
Let $M$ be a finite dimensional smooth manifold. Let $S$ be a Frölicher space with the property that there is a non-zero smooth function $C^\infty(S,\mathbb{R}) → \mathbb{R}$ with support in $C^\infty(S,(-1,1))$. Then $C^\infty(S,M)$ is a smooth manifold which is locally modelled on its kinematic tangent spaces.
Suppose, in addition, that $N$ is another finite dimensional smooth manifold and $f \colon M \to N$ a regular smooth map. Then $C^\infty(S,f) \colon C^\infty(S,M) \to C^\infty(S,N)$ is a regular smooth map.