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## Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications

#### Krishan Lal DUGGAL [1]

We introduce a pseudo Cauchy Riemann(PCR)-structure defined by a real tensor field $\bar{J}$ of type $(1, 1)$ of a real semi-Riemannian manifold $(\bar{M}, \bar{g})$ 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 framed(PF)-manifold $(M, g)$ by a real tensor field $f$ such that $f^3 = \lambda^2 f$, where $T(M)$ splits into a direct sum of two subbundles, namely $im(f)$ (with a PCR-structure) and $ker(f)$, 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$(ker(f))=1$. Contrary to the odd dimensional contact manifolds, there do exist even dimensional contact PF-manifolds. We also propose several open problems.
Semi-Riemannian metric, pseudo Cauchy Riemann manifold, pseudo framed structure, linear operator, spacetime
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Primary Language en Mathematics Research Article Orcid: 0000-0003-2967-2727Author: Krishan Lal DUGGAL (Primary Author)Institution: University of WindsorCountry: Canada University of Windsor Publication Date : January 30, 2020
 Bibtex @research article { iejg655974, journal = {International Electronic Journal of Geometry}, issn = {}, eissn = {1307-5624}, address = {}, publisher = {Kazım İLARSLAN}, year = {2020}, volume = {13}, pages = {61 - 73}, doi = {10.36890/iejg.655974}, title = {Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications}, key = {cite}, author = {DUGGAL, Krishan Lal} } APA DUGGAL, K . (2020). Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications. International Electronic Journal of Geometry , 13 (1) , 61-73 . DOI: 10.36890/iejg.655974 MLA DUGGAL, K . "Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications". International Electronic Journal of Geometry 13 (2020 ): 61-73 Chicago DUGGAL, K . "Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications". International Electronic Journal of Geometry 13 (2020 ): 61-73 RIS TY - JOUR T1 - Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications AU - Krishan Lal DUGGAL Y1 - 2020 PY - 2020 N1 - doi: 10.36890/iejg.655974 DO - 10.36890/iejg.655974 T2 - International Electronic Journal of Geometry JF - Journal JO - JOR SP - 61 EP - 73 VL - 13 IS - 1 SN - -1307-5624 M3 - doi: 10.36890/iejg.655974 UR - https://doi.org/10.36890/iejg.655974 Y2 - 2020 ER - EndNote %0 International Electronic Journal of Geometry Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications %A Krishan Lal DUGGAL %T Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications %D 2020 %J International Electronic Journal of Geometry %P -1307-5624 %V 13 %N 1 %R doi: 10.36890/iejg.655974 %U 10.36890/iejg.655974 ISNAD DUGGAL, Krishan Lal . "Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications". International Electronic Journal of Geometry 13 / 1 (January 2020): 61-73 . https://doi.org/10.36890/iejg.655974 AMA DUGGAL K . Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications. Int. Electron. J. Geom.. 2020; 13(1): 61-73. Vancouver DUGGAL K . Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications. International Electronic Journal of Geometry. 2020; 13(1): 73-61.

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