2019/10/02 by Giovanni Canarecci, Canarecci, Giovanni
Mathematics · #53A35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1910.01164
openalex publication_date 2019/10/02 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
The purpose of this study is to analyse two related topics: the Rumin\ncohomology and the \ℍ-orientability in the Heisenberg group\n\ℍn. In the first three chapters we carefully describe the Rumin\ncohomology with particular emphasis at the second order differential operator\nD, giving examples in the cases n=1 and n=2. We also show the commutation\nbetween all Rumin differential operators and the pullback by a contact map and,\nmore generally, describe pushforward and pullback explicitly in different\nsituations. Differential forms can be used to define the notion of\norientability; indeed in the fourth chapter we define the\n\ℍ-orientability for \ℍ-regular surfaces and we prove that\n\ℍ-orientability implies standard orientability, while the opposite\nis not always true. Finally we show that, up to one point, a M "obius strip in\n\ℍ1 is a \ℍ-regular surface and we use this fact to prove\nthat there exist \ℍ-regular non-\ℍ-orientable surfaces, at\nleast in the case n=1. This opens the possibility for an analysis of\nHeisenberg currents mod 2.\n