Sur les vari�t�s lorentziennes dont le groupe conforme est essentiel

Abstract

Nous proposons de nombreuses constructions de variétés compactes lorentziennes pour lesquelles le groupe conforme ne préserve aucune mesure lisse. Ceci montre que le théorème de Ferrand-Obata ne se généralise pas au cadre lorentzien.

About Pseudo-Riemannian Lichnerowicz Conjecture

“…It turns out that such a statement is far from true, already in the Lorentzian framework, as the following result shows. Fr1]). For every pair of integers (n, g), with n ≥ 3 and g ≥ 1, the manifold obtained as the product of S 1 and the connected sum of g copies of S 1 × S n−2 can be endowed with infinitely many distinct conformal Lorentz structures, each one being essential.…”

“…Nevertheless, all examples built in [Fr1] to show Theorem 1.3 are conformally flat, hence do not provide a negative answer to the local question 1.1. Actually several results obtained for instance in [BN], [FrZ], [FrM], made a positive answer to Question 1.1 plausible in full generality.…”

Einstein Spaces with a Conformal Group

“…It is the other part of the same Lichnerowicz conjecture that a compact and pseudo-Riemannian manifold carrying an essential conformal vector field is conformally flat [16]. So far it seems that in the case of an indefinite metric no example is known of a non-homothetic conformal vector field wih a zero unless the metric is locally conformally flat.…”

Dégénerescence locale des transformations conformes pseudo-riemanniennes

“…Nous commençons par de brefs rappels géométriques sur l'espace modèle conformément plat de signature (p, q), avec p q. Pour une étude plus détaillée, nous renvoyons le lecteur à [2] ou [8].…”