Version 1
: Received: 12 March 2023 / Approved: 13 March 2023 / Online: 13 March 2023 (08:38:55 CET)
How to cite:
Razpopov, D.; Dzhelepov, G.; Dokuzova, I.R. On an Indefinite Metric on a 4-Dimensional Riemannian Manifold. Preprints2023, 2023030223. https://doi.org/10.20944/preprints202303.0223.v1
Razpopov, D.; Dzhelepov, G.; Dokuzova, I.R. On an Indefinite Metric on a 4-Dimensional Riemannian Manifold. Preprints 2023, 2023030223. https://doi.org/10.20944/preprints202303.0223.v1
Razpopov, D.; Dzhelepov, G.; Dokuzova, I.R. On an Indefinite Metric on a 4-Dimensional Riemannian Manifold. Preprints2023, 2023030223. https://doi.org/10.20944/preprints202303.0223.v1
APA Style
Razpopov, D., Dzhelepov, G., & Dokuzova, I.R. (2023). On an Indefinite Metric on a 4-Dimensional Riemannian Manifold. Preprints. https://doi.org/10.20944/preprints202303.0223.v1
Chicago/Turabian Style
Razpopov, D., Georgi Dzhelepov and Iva Rumenova Dokuzova. 2023 "On an Indefinite Metric on a 4-Dimensional Riemannian Manifold" Preprints. https://doi.org/10.20944/preprints202303.0223.v1
Abstract
Our research is in the tangent space of a point on a 4-dimensional Riemannian manifold. Besides the positive definite metric, the manifold is endowed with a tensor structure of type (1,1), whose fourth power is minus the identity. Both structures are compatible and they define an indefinite metric on the manifold. With the help of the indefinite metric we determine a circle in different 2-planes in the tangent space on the manifold, we also calculate the length and area of the circle. On a smooth closed curve such as a circle, we define a vector force field. Further, we obtain the circulation done by the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analogue of the well known Green’s formula in the Euclidean space.
Keywords
Riemannian manifold; indefinite metric tensor; length; area; Green’s formula
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
