Geometric Regularity Results on B_α,β^k-Manifolds, I: Affine Connections

2019/10/07 by Yuri Ximenes Martins, Martins, Yuri Ximenes, Rodney Josué Biezuner +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1910.03113

openalex publication_date 2019/10/07 · openalex created_date 2019/10/18 · openalex updated_date 2026/07/28

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_α,β^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 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 End(TM)-valued 1-forms is suggested.

Related

• Completeness in affine and statistical geometry
• Existence of B^k_α,β-Structures on C^k-Manifolds
• Integration of 1-forms and connections
• The affine geometry of meromorphic connections with irregular singularities
• Singular Connections on Three-Manifolds and Manifolds with Cylindrical Ends