In the present paper, we study the PNDP- manifold, which has shown to be effective in applications, particularly in the field of general relativity, by introducing a new topological approach to the emergent electric or magnetic parts associated to the Weyl tensor. The working hypothesis in this study is that the electric or magnetic parts of the Weyl tensor are the dimensions of the PNDP - manifold and the interaction that emerges as ”virtual” pointlike manifolds and as a manifold with positive/ negative dimensions results in the generation of the purely electric or purely magnetic parts of the Weyl tensor. The PNDP metric for the electric or magnetic parts of the Weyl tensor is also derived.
The generalized Lorentz transformations on the PNDP manifold are introduced. These can give rise to novel forms of gauge theories if we consider local Lorentz transformations. Also we briefly show that principal fiber bundles with PNDP manifolds as base manifolds can be constructed.
Mathematics Subject classifiction (2010): 53C25; 83C99; 53A45; 53B50
In this paper our main purpose is to discuss some techniques of higher order decomposition of well-known Cartan's first curvature tensor . Moreover, we attempted to establish few significant results that may produce vital connections between Complex Finsler Manifold and Riemannian Christoffel Symbol (Curvature Tensor). Also, by adopting the techniques of decomposition, various cases and conditions have been developed.
In the present manuscript, we endeavour to review and develop the black hole solutions in general relativity. We emphasize here the Schwarzschild solution in Einstein’s field equation, which describes the gravitational field outside a spherical mass. The paper aims to obtain certain results, including the description of the Einstein field equation with stationary and static solutions and components of the metric that turns out to be
time independent, some experiments on the Schwarzschild - Penrose diagram, the Kerr-Newman solution for rotating black holes, and the Reissner- Nordstrom solution for static and charged black holes.
There is a classical fact conjectured by Albert Einstein, that the presence of matter causes the curvature of space-time. However, even a vacant space-time can have a non-zero Weyl's curvature. For instance, such a condition can be found near black holes and in the zones where gravitation waves radiate. Getting inspirations from such a fabulous classical fact, authors have attempted to describe the purely differential geometric behaviour of Weyl's-Gauge invariant conceptions concerning to 4-dimensional structured cosmos. Under the well known Ricci flow (R.F.) techniques, various Weylian configurations have been evolved as heat diffusion equations, which can pave the way for new consequencies in relativity theory and cosmology.
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