Detailed Information on Publication Record

FERRAIOLI, Diego Catalano and Michal MARVAN. The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors. Annali di Matematica Pura ed Applicata. HEIDELBERG: SPRINGER HEIDELBERG, 2020, vol. 199, No 4, p. 1343-1380. ISSN 0373-3114. Available from: https://dx.doi.org/10.1007/s10231-019-00924-y.

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Basic information
Original name	The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors
Authors	FERRAIOLI, Diego Catalano (380 Italy) and Michal MARVAN (203 Czech Republic, belonging to the institution).
Edition	Annali di Matematica Pura ed Applicata, HEIDELBERG, SPRINGER HEIDELBERG, 2020, 0373-3114.

Other information
Original language	English
Type of outcome	Article in a journal
Field of Study	10101 Pure mathematics
Country of publisher	Germany
Confidentiality degree	is not subject to a state or trade secret
WWW	Annali di Matematica Pura ed Applicata
RIV identification code	RIV/47813059:19610/20:A0000066
Organization unit	Mathematical Institute in Opava
Doi	http://dx.doi.org/10.1007/s10231-019-00924-y
UT WoS	000494394800001
Keywords in English	Differential invariants; Metric equivalence problem; Kundu class
Tags	MÚ
Tags	International impact, Reviewed
Links	GBP201/12/G028, research and development project.
Changed by	Changed by: Mgr. Aleš Ryšavý, učo 28000. Changed: 19/3/2021 12:29.

Abstract
We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the two-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants C rho and lC are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterize the Van den Bergh metric. To understand the non-generic cases, we also find all Lambda-vacuum metrics that are generic in the above sense, except that either C rho or lC is zero. In this way we extend the Kundu class to Lambda-vacuum metrics. The results of the paper can be exploited for invariant characterization of classes of metrics and for extension of the set of known solutions of the Einstein equations.

Abstract

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the two-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants C rho and lC are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterize the Van den Bergh metric. To understand the non-generic cases, we also find all Lambda-vacuum metrics that are generic in the above sense, except that either C rho or lC is zero. In this way we extend the Kundu class to Lambda-vacuum metrics. The results of the paper can be exploited for invariant characterization of classes of metrics and for extension of the set of known solutions of the Einstein equations.