2020/03/18 by Joël Merker, Joel Merker, Pawel Nurowski +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.DG #math.GR
paper · pdf · doi:10.48550/arxiv.2003.08166
37 pages, 1 figure, intensive symbolic computations
arxiv created 2020/04/15 · arxiv updated 2020/04/16
Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider 5-dimensional para-CR structures whose Levi forms are of constant rank 1 and that are 2-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs zy=F(x, y, z, zx) & zxxx=H(x,y,z,zx,zxx), with F independent of zxx and Fzxzx ≠ 0, that are completely integrable Dx3 F=Δy H, Performing at an advanced level Cartan's method of equivalence, we determine all concerned homogeneous models, together with their symmetries: (i) zy=\tfrac14 (zx)2 & zxxx=0; (ii) zy=\tfrac14 (zx)2 & zxxx=(zxx)3; (iiia) zy=\tfrac14 (zx)b & zxxx = (2-b)\frac(zxx)2zx with zx>0 for any real b∈[1,2); (iiib) zy = f(zx) & zxxx=h(zx)(zxx)2, where the function f is determined by the implicit equation: (zx2+f(zx)2) exp ( 2b arctan\tfracbzx-f(zx)zx+bf(zx) ) = 1+b2 and where: h(zx) := ((b2-3)zx-4bf(zx))/((f(zx)-bzx)2), for any real b>0.