This article is cited in 2 papers

Specificity of Petrov classification of (anti-)self-dual zero signature metrics

L. N. Krivonosov, V. A. Luk"yanov

Nizhny Novgorod State Technical University, 24 Minin str., Nizhny Novgorod, 603950 Russia

Abstract: A.Z. Petrov divided 4-metrics of the zero signature into 6 types, which later began to be denoted I, D, O, II, N, III. However, in the case of (anti)-self-duality, the $\lambda$-matrix, on the basis of which Petrov built his classification, acquires specificity. First, the determinant of this $\lambda$-matrix has a root 0 of multiplicity at least 3. Secondly, the multiplicity of this root cannot be 5. These two circumstances lead to the fact that there are not 6, but 7 different types of metrics. A new type I$_{0}$ appears, whose characteristic number 0 has multiplicity 4. This type does not coincide with I, since for type I the multiplicity of the root 0 is three. Examples of metrics, expressed in terms of elementary functions, of all seven types are constructed.

Keywords: (anti)-self-duality, Petrov classification, Weyl tensor, Hodge operator.

UDC: 514.756

Received: 04.11.2019
Revised: 08.01.2020
Accepted: 25.03.2020

DOI: 10.26907/0021-3446-2020-9-56-67

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2020, 64:9, 50–60

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