Existence of $B_{\alpha ,\beta }^k$-Structures on $C^k$-Manifolds

Abstract

In this paper we introduce $B_{\alpha ,\beta }^k$-manifolds as generalizations of the notion of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^k$-manifolds whose transition functions ${\phi }_{ji}={\phi }_j\circ {\phi }_i^{-1}$ are such that ${\partial }^{\mu }{\phi }_{ji}\in B_{\alpha (r)}\cap C^{k-\beta (r)}$ for every $|\mu |=r$, where $B=(B_r{)}_{r\in \Gamma }$ is some sequence of presheaves of Fréchet spaces endowed with further structures, $\Gamma \subset {\mathbb{Z}}_{\ge 0}$ is some parameter set and $\alpha ,\beta$ are functions. We present embedding theorems for the presheaf category of those structural presheaves $B$. The existence problem of $B_{\alpha ,\beta }^k$-structures on $C^k$-manifolds is studied and it is proved that under certain conditions on $B$, $\alpha$ and $\beta$, the forgetful functor from $C^k$-manifolds to $B_{\alpha ,\beta }^k$-manifolds has adjoints.

Minor modifications. To appear in São Paulo Journal of Mathematical Sciences