Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications

Authors : Krishan Lal DUGGAL

Pages : 61-73

Doi:10.36890/iejg.655974

View : 46 | Download : 6

Publication Date : 2020-01-30

Article Type : Research Paper

Abstract :We introduce a pseudo Cauchy Riemanninsert ignore into journalissuearticles values(PCR);-structure defined by a real tensor field $\bar{J}$ of type $insert ignore into journalissuearticles values(1, 1);$ of a real semi-Riemannian manifold $insert ignore into journalissuearticles values(\bar{M}, \bar{g});$ such that $\bar{J}^2 = \lambda^2 I$, where $\lambda$ is a function on $\bar{M}$. We prove that, contrary to the even dimensional CR-manifolds, a PCR-manifold is not necessarily of even dimension if $\lambda$ is every where non-zero real function on $\bar{M}$, supported by two odd dimensional examples and one physical model. The metric of PCR-manifold is not severely restricted. Then, we define a pseudo framedinsert ignore into journalissuearticles values(PF);-manifold $insert ignore into journalissuearticles values(M, g);$ by a real tensor field $f$ such that $f^3 = \lambda^2 f$, where $Tinsert ignore into journalissuearticles values(M);$ splits into a direct sum of two subbundles, namely $iminsert ignore into journalissuearticles values(f);$ insert ignore into journalissuearticles values(with a PCR-structure); and $ kerinsert ignore into journalissuearticles values(f);$, supported by some mathematical and physical examples. Finally, we study a revised version of a contact manifold, called contact PF-manifold, which is a particular case of a PF-manifold where dim$insert ignore into journalissuearticles values(kerinsert ignore into journalissuearticles values(f););=1$. Contrary to the odd dimensional contact manifolds, there do exist even dimensional contact PF-manifolds. We also propose several open problems.
Keywords : Semi Riemannian metric, pseudo Cauchy Riemann manifold, pseudo framed structure, linear operator, spacetime

ORIGINAL ARTICLE URL