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

TL;DR

This paper investigates the existence and multiplicity of affine connections on $B_{eta,eta}^{k}$-manifolds, establishing conditions based on structural presheaves and analyzing the space of such connections.

Contribution

It provides new existence criteria for regular affine connections on $B_{eta,eta}^{k}$-manifolds and characterizes the structure of the space of these connections.

Findings

Existence of regular affine connections characterized by properties of the structural presheaf B.

The space of regular affine connections forms an affine space modeled on regular End(TM)-valued 1-forms.

The set of regular connections is dense and open in the space of all End(TM)-valued 1-forms under a suitable topology.

Abstract

In this paper we consider existence and multiplicity results concerning affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs, following the regularity theory introduced in arXiv:1908.04442. We show that if $M$ admits a $B_{\alpha,\beta}^{k}$-structure, then the existence of such regular connections can be established in terms of properties of the structural presheaf $B$. In other words, we propose a solution to the existence problem in this setting. With regard to the multiplicity problem, we show that the space of regular affine connections is an affine space of the space of regular $\operatorname{End}(TM)$-valued 1-forms, and that if two regular connections are locally additively different, then they are actually locally different. The existence of a topology in which the space of regular connections is a nonempty open dense subset of the space of all regular $\operatorname{End}(TM)$-valued 1-forms is suggested.