Date of Award
Mathematics and Statistics
CC BY-NC-ND 4.0
Recently, there has been a keen interest of showing an interplay between definite and indefinite (in particular, Lorentzian) Riemannian geometries Flaherty (1976), Duggal (1978, 86), Beem and Ehrlich (1981), O'Neill (1983) etc . The objective of my dissertation is to present a few fresh ideas on this fruitful relationship, in reference to some applications in Relativity and Hydrodynamics. Our working spaces are the Cauchy-Riemann CR -submanifolds of a Hermitian manifold, introduced by Bejancu (1978). This choice is motivated by the fact that a CR-submanifold can be Lorentzian as opposed to a Hermitian manifold which, according to Flaherty (1976), cannot have Lorentzian signature. First part of the thesis is devoted to the characterization of Ricci-flat, Einstein and conformally flat space-times; followed by a few solutions of the Einstein Field equations. We have also shown the existence of an orthogonally transitive abelian isometry group which leads to the study on Killing Horizon Carter (1969) . In the second part, we present a few fresh ideas on the mutual interplay between the CR-structure and physical space-time with respect to the conformal geometry and its applications to hydrodynamics. The conformal geometry is further related with a symmetry property called Conformal Collineation, which has the prospect of potential physical applications. In this respect, we have studied locally symmetric manifolds and classified the shape operator of pseudo-Einstein hypersurfaces in conformally flat space. Our work provides some more information on a recent result on singularity theorems Beem and Ehrlich (1985) . Finally, we improve a recent result of Herrera et al (1985) who showed that stiff equation of state is singled out if a special conformal motion is orthogonal to the 4-velocity of an isotropic fluid. We have avoided the stiff state by using special conformal collineation and generated some new solutions of isotropic/anisotropic fluids. Source: Dissertation Abstracts International, Volume: 47-09, Section: B, page: 3804. Thesis (Ph.D.)--University of Windsor (Canada), 1986.
SHARMA, RAMESH., "CAUCHY-RIEMANN (CR)- SUBMANIFOLDS OF SEMI-RIEMANNIAN MANIFOLDS WITH APPLICATIONS TO RELATIVITY AND HYDRODYNAMICS." (1986). Electronic Theses and Dissertations. 1374.