Left-invariant Lorentzian Einstein metrics on nilpotent Lie groups.

Résumé

This thesis falls under the theme of pseudo-Riemannian geometry in the setting of Lie groups. Its purpose it to present a number of results on left-invariant Einstein Lorentzian metrics on nilpotent Lie groups as an extension of the well-known classical Riemannian case. The content of this thesis fits nicely in the subject literature since most of its results complete previous works that were initiated by many authors, some of which are even generalizations of earlier studies into broader contexts. The general outline can be divided into two major parts: The first part is concerned with the preliminaries of our study of Lorentzian left-invariant Einstein metrics on nilpotent Lie groups. The main theorem states that if the center of such a Lie group is degenerate then it must be Ricci-flat and its Lie algebra can be obtained by the double extension process from an abelian Euclidean Lie algebra. We also show that all nilpotent Lie groups up to dimension 5 endowed with a Lorentzian Einstein left-invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. The second part can be seen as a starting point for the study of Einstein Lorentzian nilpotent Lie groups with non-degenerate center as it carries over the machinery previously developed in order to treat the case of 3-step nilpotent Lie groups. The principal theorem of this part is the classification of all Einstein Lorentzian 3-step nilpotent Lie groups with 1-dimensional non-degenerate center, its proof, while long and somewhat difficult, gives insight into many different properties and aspects that were not apparent before, and the techniques used for the proof seem promising for a future inspection.

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