The Levi-Civita connections of Lorentzian manifolds with prescribed optical geometries

Dmitri V. Alekseevsky; Masoud Ganji; Gerd Schmalz; Andrea Spiro

doi:10.56754/0719-0646.2602.239

DOI:

https://doi.org/10.56754/0719-0646.2602.239

Abstract

We explicitly derive the Christoffel symbols in terms of adapted frame fields for the Levi-Civita connection of a Lorentzian \(n\)-manifold \((M, g)\), equipped with a prescribed optical geometry of Kähler-Sasaki type. The formulas found in this paper have several important applications, such as determining the geometric invariants of Lorentzian manifolds with prescribed optical geometries or solving curvature constraints.

Keywords

Levi-Civita connection , optical geometry , congruence of shearfree geodesics , Sasaki manifolds

Mathematics Subject Classification:

53B30 , 83C05

Dmitri V. Alekseevsky Institute for Information Transmission Problems, B. Karetny per. 19, 127051, Moscow, Russia - University of Hradec Králové, Faculty of Science, Rokitanského 62, 500 03 Hradec Králové, Czech Republic. https://orcid.org/0000-0002-6622-7975
Masoud Ganji School of Science and Technology University of New England, Armidale NSW 2351, Australia. https://orcid.org/0000-0001-6568-6558
Gerd Schmalz School of Science and Technology University of New England, Armidale NSW 2351, Australia. https://orcid.org/0000-0002-6141-9329
Andrea Spiro Scuola di Scienze e Tecnologie, Università di Camerino Via Madonna delle Carceri I-62032 Camerino (Macerata), Italy. https://orcid.org/0000-0001-9449-8687

Pages: 239–258
Date Published: 2024-07-09
Vol. 26 No. 2 (2024)

D. V. Alekseevsky, M. Ganji, and G. Schmalz, “CR-geometry and shearfree Lorentzian geometry,” in Geometric complex analysis, ser. Springer Proc. Math. Stat. Springer, Singapore, 2018, vol. 246, pp. 11–22.

D. V. Alekseevsky, M. Ganji, G. Schmalz, and A. Spiro, “Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry,” Differential Geom. Appl., vol. 75, 2021, Art. ID 101724, doi: 10.1016/j.difgeo.2021.101724.

A. M. Awad and A. Chamblin, “A bestiary of higher-dimensional Taub-NUT-AdS space-times,” Classical Quantum Gravity, vol. 19, no. 8, pp. 2051–2061, 2002, doi: 10.1088/0264- 9381/19/8/301.

A. L. Besse, Einstein manifolds, ser. Classics in Mathematics. Springer-Verlag, Berlin, 2008, reprint of the 1987 edition.

A. Fino, T. Leistner, and A. Taghavi-Chabert, “Optical geometries,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2023, doi: 10.2422/2036-2145.202010_050. Accepted for publication.

M. Ganji, C. Giannotti, A. Spiro, and G. Schmalz, “Einstein manifolds with optical geometries of Kerr type,” 2024, arXiv:2405.14760.

C. D. Hill, J. Lewandowski, and P. Nurowski, “Einstein’s equations and the embedding of 3-dimensional CR manifolds,” Indiana Univ. Math. J., vol. 57, no. 7, pp. 3131–3176, 2008, doi: 10.1512/iumj.2008.57.3473.

L. P. Hughston and L. J. Mason, “A generalised Kerr-Robinson theorem,” Classical Quantum Gravity, vol. 5, no. 2, pp. 275–285, 1988.

R. P. Kerr, “Scalar invariants and groups of motions in a four-dimensional Einstein space,” J. Math. Mech., vol. 12, pp. 33–54, 1963.

R. P. Kerr, “Scalar invariants and groups of motions in a Vn with positive definite metric tensor,” Tensor (N.S.), vol. 12, pp. 74–83, 1962.

B. Kruglikov and E. Schneider, “Differential invariants of Kundt spacetimes,” Classical Quantum Gravity, vol. 38, no. 19, 2021, Art. ID 195017, doi: 10.1088/1361-6382/abff9c.

F. Podestà and A. Spiro, “Introduzione ai gruppi di trasformazione,” 1996, preprint Series of the Department V. Volterra of the University of Ancona.

I. Robinson, “Null electromagnetic fields,” J. Mathematical Phys., vol. 2, pp. 290–291, 1961, doi: 10.1063/1.1703712.

I. Robinson and A. Trautman, “Conformal geometry of flows in n dimensions,” J. Math. Phys., vol. 24, no. 6, pp. 1425–1429, 1983, doi: 10.1063/1.525878.

I. Robinson and A. Trautman, “Optical geometry,” in New Theories in Physics: Proceedings of the XI Warsaw Symposium on Elementary Particle Physics. Teaneck, NJ, USA: World Scientific Publishing Co., 1989, pp. 454–497.

S. Sternberg, Lectures on differential geometry, 2nd ed. Chelsea Publishing Co., New York, 1983.