2024/07/18 by Fernandez, Javier, Kordon, Francisco
#53B15 #53C05 (Primary) 37J06 #70G45 (Secondary) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2407.13614
In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle ϕ:Q→ M may be used to split TQ into horizontal and vertical subbundles, a discrete connection may be used to split Q× Q into horizontal and vertical submanifolds. All discrete connections induce a connection on the same principal bundle via a process known as the Lie or derivative functor. The Integration Problem consists of describing, for a principal connection A, the set of all discrete connections whose associated connection is A. Our first result is that for flat principal connections, the Integration Problem has a unique solution among the flat discrete connections. More broadly, under a fairly mild condition on the structure group G of the principal bundle ϕ, we prove that the existence part of the Integration Problem has a solution that needs not be unique. Last, we see that, when G is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures.