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

@article{Martins2020GeometricRR, title={Geometric Regularity Results on \$B\_\{\alpha,\beta\}^\{k\}\$-Manifolds, I: Affine Connections}, author={Yuri Ximenes Martins and Rodney Josu'e Biezuner}, journal={arXiv: Differential Geometry}, year={2020} }

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…

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