2020/02/10 by Elena Mirela Babalic, Babalic, Elena Mirela, Stefan Berceanu +1
Mathematics · #32F45 #53C30 #53C55 #81R30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.DG #msc:32F45 #msc:53C30 #msc:53C55 #msc:81R30
paper · pdf · doi:10.48550/arxiv.2002.04452
26 pages, Latex, amsart, AMS fonts; the abstract and the introduction are improved, some typos are eliminated, more references are added; arXiv admin note: text overlap with arXiv:1903.10721
openalex publication_date 2020/02/10 · arxiv created 2020/05/21 · arxiv updated 2020/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The real Jacobi group GJ1(ℝ)=\rm SL(2,ℝ)\ltimes \rm H1, where \rm H1 denotes the 3-dimensional Heisenberg group, is parametrized by the S-coordinates (x,y,θ,p,q,κ). We show that the parameter η that appears passing from Perelomov's un-normalized coherent state vector based on the Siegel--Jacobi disk DJ1 to the normalized one is η=q+\rmi p. The two-parameter invariant metric on the Siegel--Jacobi upper half-plane XJ1=\fracGJ1(\R)\rmSO(2)×ℝ is expressed in the variables (x,y,\rmRe~η,\rmIm~η). It is proved that the five dimensional manifold XJ1=\fracGJ1(\R)\rmSO(2)\approxXJ1×ℝ, called extended Siegel--Jacobi upper half-plane, is a reductive, non-symmetric, non-naturally reductive manifold with respect to the three-parameter metric invariant to the action of GJ1(ℝ), and its geodesic vectors are determined.