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

DOI10.31392/MFAT-NPU26_3.2021.05zbMATH Open1513.53052arXiv1910.03113MaRDI QIDQ6326717

Yuri Ximenes Martins, Rodney Josué Biezuner

Publication date: 7 October 2019

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 $$-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 $operatornameEnd(TM)$-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 $operatornameEnd(TM)$-valued 1-forms is suggested.

Mathematics Subject Classification ID

Differentiable manifolds, foundations (58A05) Categories, functors in functional analysis (46M15) Connections (general theory) (53C05)

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