2020/06/07 by Fernand Pelletier, Pelletier, Fernand
Mathematics · #53D35 #55P35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Limit (mathematics) #Mathematical analysis #Mathematics #Pure mathematics #Sequence (biology) #Symplectic Geometry (math.SG) #Symplectic geometry #Tangent bundle #Tangent space #math.DG #math.SG #msc:53D35 #msc:55P35
paper · pdf · doi:10.48550/arxiv.2006.04104
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2020/06/07 · arxiv created 2020/07/21 · arxiv updated 2020/07/22 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/08
Given a projective sequence of Banach bundles, each one provided with a of\nweak symplectic form, we look for conditions under which, the corresponding\nsequence of weak symplectic forms gives rise to weak symplectic form on the\nprojective limit bundle. Then we apply this results to the tangent bundle of a\nprojective limit of Banach manifolds. This naturally leads to ask about\nconditions under which the Darboux Theorem is also true on the projective limit\nof Banach manifolds. We will give some necessary and some sufficient conditions\nso that such a result is true. Then we discuss why, in general, the Moser's\nmethod can not work on projective limit of Banach weak symplectic Banach\nmanifolds without very strong conditions like Kumar 's results ([17]). In\nparticular we give an example of a projective sequence of weak symplectic\nBanach manifolds on which the Darboux Theorem is true on each manifold, but is\nnot true on the projective limit of these manifolds.\n