Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications

Year 2020, , 61 - 73, 30.01.2020

Krishan Lal Duggal

https://doi.org/10.36890/iejg.655974

Cited By: 1

Abstract

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.

Keywords

Semi-Riemannian metric, pseudo Cauchy Riemann manifold, pseudo framed structure, linear operator, spacetime

Supporting Institution

University of Windsor

References

• [1] Beem, J. K., Ehrlich, P. E.: Global Lorentzian Geometry. Marcel Dekker Inc. New York (1981). Second Edition (with Easley, K. L.) (1996).
• [2] Bejancu, A.: Geometry of CR-submanifolds. D. Reidel Publishing Company. Boston (1986).
• [3] Blair, D. E.: Contact manifolds in Riemannian geometry. Lecture notes in Math. Springer-Verlag. Berlin (1976).
• [4] Blair, D. E.: Geometry of manifolds with structure group U(n) × O(s). J. Diff. Geometry 4, 155–167 (1970).
• [5] Duggal, K. L.: Spacetime manifolds and contact structures. Int. J. Math. & Math. Sci. 13, 545–554 (1990).
• [6] Duggal K. L.: Warped product of lightlike manifolds. Nonlinear Anal. 47, 3061-3072 (2001).
• [7] Duggal, K. L., Jin, D. H.: Null Curves and Hypersurfaces of Semi-Riemannian Manifolds. World Scientific (2007).
• [8] Duggal, K. L., Sahin, B.: Differential geometry of lightlike submanifolds. Birkhäuser (2010).
• [9] Duggal, K. L., Sharma, R.: Symmetries of spacetimes and Riemannian manifolds. Kluwer Academic Publishers (1999).
• [10] Eliopoulos, H.: On the general theory of differentiable manifolds with almost tangent structure. Canad. Math. Bull. 8, 721–748 (1965).
• [11] Flaherty, E. T.: Hermitian and Kählerian Geometry in Relativity. Lecture Notes in Physics. Springer-Verlag. Berlin (1976).
• [12] Gray, A., Hervella, L. M.: The sixteen classes of almost Hermitian manifolds. Ann. Mt. Pura Appli. 123, 35–58 (1980).
• [13] Kaehler, E.: Über eine bemerkenswerte Hermitische metrik. Abh. Math. Sem. Hamburg 9, 173-186 (1933).
• [14] Legrand, G.: Sur les variéteś à structure de presque produit complexe. C.R. Acad. Sci. Paris 242, 335–337 (1956).
• [15] Middleton. C. A., Stanley. E.: Anisotropic evolution of 5D Friedmann-Robertson-Walker spacetime. Phys. Rev. D84 085013, (2011).
• [16] Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 65, 391–404 (1957).
• [17] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press. New York (1983).
• [18] Penrose, R.: Physical spacetime and non realizable CR-structure. Proc. of symposia in Pure Math. 39, 401–422 (1983).
• [19] Walker, A. G.: Completely symmetric spaces. J. Lond. Math. Soc. 19, 219–226 (1944).
• [20] Yano, K.: On a structure defined by a tensor field of type (1,1) satisfying $f^3 + f = 0$. Tensor N.S. 14, 99–109 (1963)