Geometric Regularity Results on $B_{\alpha,\beta}^{k}$-Manifolds, I: Affine Connections.

In this paper we consider the existence problem of affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs. We show that if $M$ admits a suitable subatlas, meaning a $\mathcal{B}_{\alpha,\beta}^{k}$-structure for a certain presheaf of Fr\'echet spaces $B$ and for certain functions $\alpha$ and $\beta$, then the existence of such regular connections can be established. It is also proved that if the $\mathcal{B}_{\alpha,\beta}^{k}$-structure is actually nice (in the sense of arXiv:1908.04442), then a multiplicity result can also be obtained by means of Thom's transversality arguments.