Ricci Flat 4-Metrics with Bidimensional Null Orbits
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Ricci Flat 4-Metrics with Bidimensional Null Orbits Part I. General Aspects and Nonabelian Case D. Catalano Ferraioli · A. M. Vinogradov
Received: 9 November 2004 / Accepted: 8 February 2006 / Published online: 4 July 2006 © Springer Science+Business Media B.V. 2006
Abstract Pseudo-Riemannian 4-metrics with bidimensional null Killing orbits are studied. Both Lorentzian and Kleinian (or neutral) cases, are treated simultaneously. Under the assumption that the distribution orthogonal to the orbits is completely integrable a complete exact description of Ricci flat metrics admitting a bidimensional nonabelian Killing algebra is found. Mathematics Subject Classification (2000) primary 53B30, 53C25, 53C15 · secondary 53B50, 83C35 Key words Einstein metrics · Ricci flat metrics · Killing algebra · null orbits · Kleinian metrics · Lorentzian metrics
1. Introduction In this and the forthcoming paper [4] we give an exact description of Ricci flat 4-metrics g under assumptions that (i) g admits a Killing algebra G with bidimensional leaves (orbits of G ), (ii) the distribution D⊥ orthogonal to Killing leaves is Frobenius (completely integrable), (iii) g degenerates when restricted to any Killing leaf.
D. Catalano Ferraioli (B) Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy e-mail: [email protected] A. M. Vinogradov Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy A. M. Vinogradov e-mail: [email protected]
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Acta Appl Math (2006) 92: 209–239
Condition (i) implies that dim G ≥ 2 and, moreover, that G contains a (not necessarily proper) 2-dimensional subalgebra. There exists only one, up to isomorphism, 2-dimensional nonabelian Lie algebra G2 and the case in which G ⊇ G2 is studied in this paper. The case G ⊇ A2 , A2 being an Abelian 2-dimensional algebra, is the subject of the forthcoming paper [4]. Condition (iii) subjects such a metric to be either of signature (−+++) or (−−++). Due to their importance in general relativity Lorentzian, i.e., of signature (−+++), Ricci flat metrics were intensively studied for decades. In particular, almost exhaustive classification of this kind of metrics admitting at least two Killing fields was done. See, for instance, Petrov [13] and Stephani et al. [18]. On the contrary, relatively few results were obtained for Ricci flat Kleinian (or neutral), i.e., of signature (−−++), metrics. We stress from the very beginning that while new results of this and forthcoming paper [4] concern mainly Kleinian metrics our approach makes no distinction between these two cases. Ricci flat manifolds of Kleinian signature possess a number of interesting geometrical properties and undoubtedly deserve attention in their own right. Some topological aspects of these manifolds were studied for the first time in Matsushita [9, 10] and subsequently in Law [12]. In recent years geometry of these manifolds has seen a revival of interest. In part, this is due to the emergence of some new applications
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