摘要
本文从光滑流形M上的(n-1,k)-狄拉克结构出发研究南部—泊松几何的性质.首先研究高阶Courant代数胚上自同构群和无穷小,证明无穷小Courant自同构的可积性.在光滑态射φ:N→M与M上(n-1,k)-狄拉克结构的锚映射横截的条件下,得到M上(n-1,k)-狄拉克结构的拉回为N上的(n-1,k)-狄拉克结构.其次给出南部—泊松结构的图是(n-1,n-2)-狄拉克结构,得到南部—泊松结构的单参数簇在规范变换下与一簇闭的n-辛形式有关.当φ:N→M作为余(n-1)-辛子流形的浸入映射时,M上南部—泊松结构的拉回为N上的南部—泊松结构.最后讨论了M上的(n-1,0)-狄拉克结构可积分为(n-1)-预辛群胚的问题.在映射Π:M→M/H下,其对应的(n-1,0)-狄拉克结构分别是F和E.如果E可积分为(n-1)-预辛群胚(g,ω),则存在唯一的ω,使F对应积分为(n-1)-预辛群胚(g,ω).
In this paper,we show that the(n-1,k)-Dirac structures on smooth manifold M and the propositions of Nambu-Poisson geometry.First,we research Courant algebroid's automorphisms and the infinitesimal,and prove the integrability of infinitesimal Courant automorphisms.We pull back a(n-1,k)-Dirac structure under a smooth mapφ:N→M.Such pull back operations are defined by(n-1,k)-Dirac structures,under transversality assumptions.Second,we prove that the graph of Nambu-Poisson structure is the(n-1,n-2)-Dirac structure,which is the one parameter family of Nambu-Poisson structures related to the family closed nsymplectic forms under the gauge transformation.The Nambu-Poisson structure on M is pulled back to the Nambu-Poisson structure on N by the immersion mapφ:N→M as an(n-1)-cosymplectic submanifold.Finally,we discuss that the(n-1,0)-Dirac structure on M can be integrated to(n-1)-presymplectic groupoid.By the map q:M→M/H,the corresponding(n-1,0)-Dirac structures are F and E.If E can be integrated by(n-1)-presymplectic groupoid(g,ω),then there exists uniqueωso that F can be integrated by(n-1)-presymplectic groupoid(g,ω).
作者
毕艳会
李佳
BI Yanhui;LI Jia(Center for Mathematical Sciences,College of Mathematics and Information Science,Nanchang Hangkong University,Jiangxi,330068,P.R.China)
出处
《数学进展》
CSCD
北大核心
2023年第5期867-882,共16页
Advances in Mathematics（China）
基金
国家自然科学基金(Nos.11961049,11601219)
