An oft-used example of a regular map of finite-dimensional smooth manifolds is a submersion. We have the well-known result that the pullback of a submersion exists and is a submersion. For Fréchet manifolds, one definition of a submersion $f\colon M\to N$ is a map such that every induced map on tangent spaces $T_m M \to T_f(m) N$ is a split map of Frechet spaces. Here surjectivity is not enough. We know that the pullback of a submersion in this sense exists and is a submersion (cf. Hamilton's paper "The inverse function theorem of Nash and Moser"). So we could define a surjective regular map between Fréchet manifolds as being a submersion in this sense. My first question is then:
What is the full definition of a smooth, regular map of Fréchet spaces?
‘Maximal rank’ on each tangent space is not enough, as we have seen.
But there are two things flowing on from this which I don't quite get. One is that, considering Fréchet manifolds as diffeological spaces, there is another concept that is used, namely being a subduction (e.g. Thomas Nikolaus's answer to Advantages of Diffeological Spaces over General Sheaves), which is something like ‘is a submersion on each plot’. Now the pullback of a map of Fréchet manifolds which is a subduction is not guaranteed to be a Fréchet manifold. If it was I would be very happy.
Can we say anything about whether the pullback diffeological space is a Fréchet manifold?
The second thing is that really in the case of constructing pullbacks of smooth manifolds, all we need is transversality of a pair of smooth maps with common codomain. At the Wikipedia page on transversality, versions for Banach spaces are mentioned, but no details, and clearly the step from Banach to Fréchet spaces is another step removed.
What is the correct definition of transversal maps of Fréchet manifolds, and what is the statement about existence of pullbacks of transversal maps?
