Geometric Regularity Results on B k α , β -Manifolds, I: Affine Connections

In this paper we consider the existence problem of affine connections on C-manifolds M whose coefficients are as regular as one needs. We show that if M admits a suitable subatlas, meaning a B α,β-structure for a certain presheaf of Fréchet spaces B and for certain functions α and β, then the existence of such regular connections can be established. It is also proved that if the B α,β-structure is actually nice (in the sense of [1]), then a multiplicity result can also be obtained by means of Thom’s transversality arguments.