# Submersion theorem for smooth tame Frechet manifolds

If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a regular submanifold of $M$.

Question: Is there an analogous result for maps between manifolds modeled off locally convex spaces, in particular maps between smooth tame Frechet manifolds?

Question: Let $U$ be an open set in a smooth tame Frechet space $E$ (hence $U$ is a smooth tame Frechet manifold), $F$ be a smooth tame Frechet space, and $f:U\rightarrow F$ a smooth tame map. Is there a submersion or regular level set theorem in this context?

• You'd need to take 'regular value' (EDIT: at least for dim(source) bigger than dim(target)) to mean that in a neighbourhood the map looks like a split surjection, not just surjective on tangent spaces (or even a split surjection on tangent spaces). In that case, the fibre is, I think, a regular submanifold. I'm sure this should be true for locally convex manifolds in general. – David Roberts Aug 12 '15 at 2:58
• Have you looked up Nash's implicit function theorem? – Ryan Budney Aug 12 '15 at 6:16

## 1 Answer

There are a few works which study submersions in the locally convex setting. The most extensive (and recent) is by Helge Glöckner (http://arxiv.org/abs/1502.05795). Note that the basic results about submersions and immersions still hold true in this general setting, at least if you use the right notion of sub/immersion (i.e. a map being locally a projection or injection). The problem is essentially that you cannot easily go from infinitesimal results to local ones (as this step requires an inverse function theorem). Submanifolds of co-Banach type are discussed in http://arxiv.org/abs/1305.3145. Moreover, Hamilton's 1982 treatise of the Nash-Moser inverse function theorem also contains some results about submanifolds in the tame Fréchet context. However, these are rather hidden and only worked-out in specific examples (for example, his proof that the volume-preserving diffeomorphism group is a Lie subgroup of all diffeomorphisms is essentially a regular-value type argument).