Sur l'holonomie des variétés pseudo-riemanniennes

Abstract : The three chapters, quite independent, study the pseudo-Riemannian manifolds (manifolds endowed with a nondegenerate but indefinite metric) the restricted holonomy of which is indecomposable but preserves some totally isotropic subspaces. Chapter 1. A Riemannian metric with parallel Ricci curvature is locally (globally if it is complete and simply connected) a product of Einstein manifolds. This follows from the fact that the metric is positive definite and fails in the pseudo-Riemanian case. However, using in particular the Bianchi identity, clasical properties of the holonomy and a work of Klingenberg on the pairs of symmetric bilinear forms, we show a similar result : the metrics is decomposed into a product of Einstein metrics, <> and metrics with <> (see th. 1 p. 19). Chapter 2. It restates and generalizes Klingenberg's work used in chapter 1. It classifies the pairs of reflexive bilinear forms over some field K, in finite dimension, up to conjugation by the linear group, and gives the structure of the automorphism group of the pair. In the real case, it gives a table of <> for pairs of reflexive forms, see pp. 96-100. Chapter 3. The more significant. It builds, on a certain class of pseudo-Riemannian irreducible metrics, some <> coordinates, in a sense that it gives (th. 1 p. 167). These coordinates are a tool to understand the local geometry of these metrics. In particular, they enable to parametrize the space of germs of Lorentzian metrics corresponding to the four types of possible Lorentzian holonomies given by A. Ikemakhen et L. Bérard Bergery. (see pp. 204--205 et 211).
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Mathematics [math]. Université Henri Poincaré - Nancy I, 2000. French


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Charles Boubel. Sur l'holonomie des variétés pseudo-riemanniennes. Mathematics [math]. Université Henri Poincaré - Nancy I, 2000. French. <tel-00008842>

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