Tame Fréchet submanifolds of co-Banach type (Q2353631)

\textit{R. S. Hamilton} [Bull. Am. Math. Soc., New Ser. 7, 65--222 (1982; Zbl 0499.58003)] introduced tame Fréchet spaces as Fréchet spaces with a finer analytic structure, which allow them to have a useful inverse function theorem (general Fréchet spaces do not). Accordingly, tame linear maps and tame smooth maps can be defined. A tame Fréchet manifold is a manifold with charts in a tame Fréchet space, with transition maps which are tame smooth maps. The author introduces here the notion of tame Fréchet submanifolds of co-Banach type: if \(M\) is a tame Fréchet manifold modeled in \(F\), \(N\subset M\) is a submanifold of co-Banach type if it has (adapted) charts modeled in a complemented (tame) suspace \(F_0\subset F\), which has a supplement \(B\) that is (isomorphic to) a Banach space \(B\). Moreover, \(M\) is of co-finite type if \(B\) is finite dimensional. The author proves an implicit function theorem for tame smooth maps (Theorem 4.2) based on the Nash-Moser inverse function theorem. The notion of tame regular points and values of a tame smooth map are introduced. The main result of the paper (Theorem 5.3) states the following: If \(\varphi :N_F\to N_B\) is a tame smooth map from a tame Fréchet manifold \(N_F\) (modeled in \(F\)) to a Banach manifold \(N_B\) (modeled in the Banach space \(B\)), and \(g\in N_B\) is a regular value of \(\varphi\), then \(\varphi^{-1}(g)\) is a tame Fréchet submanifold of co-Banach type. Using this result, several examples of tame Fréchet submanifolds of co-Banach or co-finite type are constructed. The author states that these results are part of a technical framework in a program to develop a theory of affine Kac-Moody symmetric spaces.