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Conformal actions of higher-rank lattices on pseudo-Riemannian manifolds Pecastaing, Vincent in Geometric and Functional Analysis (2020), 30 We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which ... [more ▼] We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group ([33]). When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of [1] and [17] in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exceptional groups. Our work is strongly inspired by the recent breakthrough of Brown, Fisher and Hurtado on Zimmer’s conjecture [7]. [less ▲] Detailed reference viewed: 73 (4 UL)The conformal group of a compact simply connected Lorentzian manifold ; Pecastaing, Vincent E-print/Working paper (2019) Detailed reference viewed: 124 (1 UL)Projective and conformal closed manifolds with a higher-rank lattice action Pecastaing, Vincent E-print/Working paper (2019) We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo ... [more ▼] We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p,q), with min(p,q)>1. In the continuity of a recent article, provided that such a structure is locally equivalent to its model X, the main question treated here is the completeness of the associated (G,X)-structure. The similarities between the model spaces of non-Lorentzian conformal geometry and projective geometry make that lots of arguments are valid for both cases, and we expose the proofs in parallel. The conclusion is that in both cases, when the real-rank is maximal, the manifold is globally equivalent to either the model space X or its double cover. [less ▲] Detailed reference viewed: 64 (0 UL)Conformal actions of real-rank 1 simple Lie groups on pseudo-Riemannian manifolds Pecastaing, Vincent in Transformation Groups (2019), 24(4), 1213-1239 Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the ... [more ▼] Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we obtain as corollary the list of semi-simple Lie groups without compact factor that can act on compact Lorentzian manifolds. We also derive consequences in CR geometry via the Fefferman fibration. [less ▲] Detailed reference viewed: 137 (7 UL)Lorentzian manifolds with a conformal action of SL(2,R) Pecastaing, Vincent in Commentarii Mathematici Helvetici (2018), 98(2), 401-439 We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the ... [more ▼] We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywhere locally conformally diffeomorphic to Minkowski space-time. When the group is non-compact and not locally isomorphic to SO(1,n), n>1, we derive global conclusions, extending a theorem of Frances and Zeghib to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of the conformal groups of compact Lorentz manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib at the end of the 1990's. [less ▲] Detailed reference viewed: 108 (2 UL)Conformal essential actions of PSL(2,R) on real-analytic compact Lorentz manifolds Pecastaing, Vincent in Geometriae Dedicata (2017), 188(1), 171-194 The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least three admitting a conformal essential action of a Lie group locally isomorphic to ... [more ▼] The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least three admitting a conformal essential action of a Lie group locally isomorphic to PSL(2,R). It is established by using a general result on local isometries of real-analytic rigid geometric structures. As corollary, we deduce the same conclusion for conformal essential actions of connected semi-simple Lie groups on real-analytic compact Lorentz manifolds. This work is a contribution to the understanding of the Lorentzian version of a question asked by A. Lichnerowicz. [less ▲] Detailed reference viewed: 55 (3 UL)On two theorems about local automorphisms of geometric structures Pecastaing, Vincent in Annales de l'Institut Fourier (2016), 66(1), 175-208 This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give suf- ficient conditions for local homogeneity in a broad class ... [more ▼] This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give suf- ficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemannian manifolds. We also revisit a strong result of Gromov which describes the structure of the orbits of local automorphisms of manifolds endowed with A- rigid structures, and give a statement and a simpler proof of this result in the setting of Cartan geometries. [less ▲] Detailed reference viewed: 92 (1 UL) |
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