Geodesic
Specificity of Petrov classification of (anti-)self-dual zero signature metrics
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2020), pp. 56-67

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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. 
@article{IVM_2020_9_a4,
     author = {L. N. Krivonosov and V. A. Luk"yanov},
     title = {Specificity of {Petrov} classification of (anti-)self-dual zero signature metrics},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {56--67},
     publisher = {mathdoc},
     number = {9},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_9_a4/}
}
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L. N. Krivonosov; V. A. Luk"yanov. Specificity of Petrov classification of (anti-)self-dual zero signature metrics. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2020), pp. 56-67. http://geodesic.mathdoc.fr/item/IVM_2020_9_a4/