Hermitian metrics with (anti-)self-dual Riemann tensor
- Authors: Krivonosov L.1, Luk'yanov V.1
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Affiliations:
- Nizhny Novgorod State Technical University
- Issue: Vol 25, No 4 (2021)
- Pages: 616-633
- Section: Differential Equations and Mathematical Physics
- Submitted: 16.06.2021
- Accepted: 12.10.2021
- Published: 30.12.2021
- URL: https://journals.eco-vector.com/1991-8615/article/view/71602
- DOI: https://doi.org/10.14498/vsgtu1867
- ID: 71602
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Abstract
Equations of (anti-)self-duality for the components of the Levi–Civita connection of the Hermitian positive definite metric (not for the Riemann tensor) are compiled. With this well-known method, a simpler system of partial differential equations is obtained, which implies the (anti-)self-duality of the Riemann tensor. This system is of the 1st order, while the (anti-)self-duality conditions of the Riemann tensor are expressed by equations of the 2nd order. However, this method can obtain only particular solutions of the (anti-)self-duality equations of the Riemann tensor. The constructed equations turned out to be significantly different in the self-dual and anti-self-dual cases. In the case of self-duality, the equations are divided into three classes, for each of which a general solution is found. In the anti-self-dual case, we did not find the general solution, but gave two series of particular solutions. The connection between our solutions and Kähler metrics is shown. In the case of the (anti-)self-duality of the Levi–Civita connection for the Hermitian metric, a general form of parallel almost complex metric-preserving structures is obtained. These structures are all torsion free. For an arbitrary positive definite 4-metric, a general form of almost complex structures preserving this metric is found.
About the authors
Leonid Krivonosov
Nizhny Novgorod State Technical University
Email: l.n.krivonosov@gmail.com
ORCID iD: 0000-0002-3533-9595
SPIN-code: 3809-5438
Scopus Author ID: 56116955100
http://www.mathnet.ru/person36147
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematics
24, Minina st., Nizhnii Novgorod, 603600, Russian FederationVyacheslav Luk'yanov
Nizhny Novgorod State Technical University
Author for correspondence.
Email: oxyzt@ya.ru
ORCID iD: 0000-0002-7294-0232
SPIN-code: 8711-9605
Scopus Author ID: 56117934500
http://www.mathnet.ru/person44358
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematics
24, Minina st., Nizhnii Novgorod, 603600, Russian FederationReferences
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