Geodesic Mappings of Conformally Flat and Three-Dimensional Pseudo-Riemannian Spaces
DOI:
https://doi.org/10.58649/r35csv10Keywords:
pseudo-Riemannian space, geodesic mapping, conformally flat space, Einstein tensor, Sinyukov equationsAbstract
Geodesic mappings constitute an important topic in differential geometry and play a significant role in mathematical physics, especially in the geometric study of gravitational and other field theories. In this paper, we investigate geodesic mappings of conformally flat (pseudo)-Riemannian spaces of dimension n > 3 and of arbitrary three-dimensional (pseudo)-Riemannian spaces.
Using a lemma due to Moldobayev concerning the structure of solutions of the Sinyukov equations, we show that the fundamental tensor associated with a geodesic mapping can be represented as a linear combination of the metric tensor and the Ricci tensor. By means of the Einstein tensor, we derive a new intrinsic characterization of conformally flat spaces and three-dimensional spaces admitting nontrivial geodesic mappings.
In particular, we obtain a recurrent-type differential condition for the Einstein tensor and show that this condition is necessary and sufficient under natural gradient assumptions. The obtained results provide internal criteria for the existence of nontrivial geodesic mappings and clarify the geometric structure of such spaces.
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No datasets were generated or analysed during the current study.
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Copyright (c) 2026 Josef Mikeš, Irene Hinterleitner, Dz. Moldobayev (Author)
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Articles published in Journal of Modern Mathematical Studies are distributed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Authors retain copyright and grant the journal the right of first publication.
