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Abdelghani Zeghib - One of the best experts on this subject based on the ideXlab platform.
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Simply connected indefinite homogeneous spaces of finite volume
2018Co-Authors: Oliver Baues, Wolfgang Globke, Abdelghani ZeghibAbstract:Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with Isometry Group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie Group acting transitively on $M$. For metric index less than three, we find that the Isometry Group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decomposition into three distinct types of metric Lie algebras.
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on the Isometry Group of lorentz manifolds
2012Co-Authors: Leandro Lichtenfelz, Paolo Piccione, Abdelghani ZeghibAbstract:We will first review a construction in [10] to establish the Lie Group structure of the Group of isometries of a semi-Riemannian manifold. The problem is cast in the language of G-structures. In the second part of this chapter, we will review some recent results on the classification of Groups acting isometrically on compact Lorentz manifolds and on the geometry of compact manifolds whose Isometry Group is non compact.
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on the Isometry Group and the geometric structure of compact stationary lorentzian manifolds
arXiv: Differential Geometry, 2010Co-Authors: Paolo Piccione, Abdelghani ZeghibAbstract:We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose Isometry Group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike) manifold.
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the identity component of the Isometry Group of a compact lorentz manifold
Duke Mathematical Journal, 1998Co-Authors: Abdelghani ZeghibAbstract:This result may be compared with a Theorem of E. Ghys [Ghy] (see also [Bel]), asserting a similar conclusion, but assuming that M has dimension 3, and that the action is just volume preserving and locally free. The statement there, is that the action of AG may be extended to an action of a finite cover of PSL(2,R), or to an action of the solvable 3-dimensional Lie Group SOL. Here we have another motivation. We want to understand the structure of Lie Groups acting isometrically on compact Lorentz manifold. The first results in the subject are due to [Zim] and [Gro]. A “final” result is due to [A-S] and [Zeg1], independently. Necessary and sufficient conditions were given in order that a Lie Group acts isometrically (and locally faithfully) on a compact Lorentz manifold. Note however, that if a Group acts in such a fashion, then its subGroups also act in the same way. For instance, all known examples of isometric actions of AG are obtained by viewing it as a subGroup of SL(2,R). So a natural question is: what are the maximal (connected) Lie Groups acting isometrically on a compact Lorentz manifold? Equivalently:
Yi Zhou - One of the best experts on this subject based on the ideXlab platform.
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global regularity for einstein klein gordon system with u 1 r Isometry Group i
Chinese Annals of Mathematics Series B, 2020Co-Authors: Haoyang Chen, Yi ZhouAbstract:This is the first of the two papers devoted to the study of global regularity of the 3 + 1 dimensional Einstein-Klein-Gordon system with a U(1) × ℝ Isometry Group. In this first part, the authors reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2 + 1 dimensional system. Then, the authors will give energy estimates and construct the null coordinate system, under which the authors finally show that the first possible singularity can only occur at the axis.
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global regularity for einstein klein gordon system with u 1 times mathbb r Isometry Group i
arXiv: Analysis of PDEs, 2019Co-Authors: Haoyang Chen, Yi ZhouAbstract:This is the first of the two papers devoted to the study of global regularity of the 3+1 dimensional Einstein-Klein-Gordon system with a $U(1)\times \mathbb{R}$ Isometry Group. In this first part, we reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2+1 dimensional system. Then, we will give energy estimates and construct the null coordinate system, under which we finally show that the first possible singularity can only occur at the axis.
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global regularity for einstein klein gordon system with u 1 times mathbb r Isometry Group ii
arXiv: Analysis of PDEs, 2019Co-Authors: Haoyang Chen, Yi ZhouAbstract:This paper is devoted to the study of the global existence of smooth solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1) \times \mathbb{R}$ Isometry Group for a class of regular Cauchy data. In our first paper \cite{chen}, we reduce the Einstein equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof for the global regularity for the 2+1 dimensional system. Firstly, we show the non-concentration of the energy near the first possible singularity. Then, we prove that the global regularity holds for initial data with small energy.
Debashish Goswami - One of the best experts on this subject based on the ideXlab platform.
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Existence of quantum Isometry Group for a class of compact metric spaces
2012Co-Authors: Debashish GoswamiAbstract:We formulate a definition of isometric action of a compact quantum Group (CQG) on a compact metric space, generalizing Banica’s definition for finite metric spaces, and show that any CQG action on a compact Riemannian manifold which is isometric in the geometric sense of [12] automatically satisfies the Isometry condition of the present article. We also prove for certain special class of metric measure spaces the existence of the universal object in the category of those compact quantum Groups which act isometrically and in a measure-preserving way.
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quantum Isometry Group for spectral triples with real structure
Symmetry Integrability and Geometry-methods and Applications, 2010Co-Authors: Debashish GoswamiAbstract:Given a spectral triple of compact type with a real structure in the sense of (Dabrowski L., J. Geom. Phys. 56 (2006), 86-107) (which is a modification of Connes' original definition to accommodate examples coming from quantum Group theory) and refe- rences therein, we prove that there is always a universal object in the category of compact quantum Group acting by orientation preserving isometries (in the sense of (Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572)) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum Isometry Group in the context of real spectral triples without fixing a choice of 'volume form' as in (Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572).
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quantum Isometry Groups examples and computations
Communications in Mathematical Physics, 2009Co-Authors: Jyotishman Bhowmick, Debashish GoswamiAbstract:In this follow-up of [4], where the quantum Isometry Group of a noncommutative manifold has been defined, we explicitly compute such quantum Groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum Isometry Group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum Isometry Group of the original (undeformed) manifold.
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quantum Group of isometries in classical and noncommutative geometry
arXiv: Quantum Algebra, 2007Co-Authors: Debashish GoswamiAbstract:We formulate a quantum generalization of the notion of the Group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum Group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum Isometry Group) in the category of compact quantum Groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum Isometry Group with the universal object in a bigger category, namely the category of `quantum families of smooth isometries', defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum Isometry Group. We give explicit description of quantum Isometry Groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in \cite{hajac} as the universal quantum Group of holomorphic isometries of the noncommutative torus.
Haoyang Chen - One of the best experts on this subject based on the ideXlab platform.
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global regularity for einstein klein gordon system with u 1 r Isometry Group i
Chinese Annals of Mathematics Series B, 2020Co-Authors: Haoyang Chen, Yi ZhouAbstract:This is the first of the two papers devoted to the study of global regularity of the 3 + 1 dimensional Einstein-Klein-Gordon system with a U(1) × ℝ Isometry Group. In this first part, the authors reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2 + 1 dimensional system. Then, the authors will give energy estimates and construct the null coordinate system, under which the authors finally show that the first possible singularity can only occur at the axis.
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global regularity for einstein klein gordon system with u 1 times mathbb r Isometry Group i
arXiv: Analysis of PDEs, 2019Co-Authors: Haoyang Chen, Yi ZhouAbstract:This is the first of the two papers devoted to the study of global regularity of the 3+1 dimensional Einstein-Klein-Gordon system with a $U(1)\times \mathbb{R}$ Isometry Group. In this first part, we reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2+1 dimensional system. Then, we will give energy estimates and construct the null coordinate system, under which we finally show that the first possible singularity can only occur at the axis.
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global regularity for einstein klein gordon system with u 1 times mathbb r Isometry Group ii
arXiv: Analysis of PDEs, 2019Co-Authors: Haoyang Chen, Yi ZhouAbstract:This paper is devoted to the study of the global existence of smooth solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1) \times \mathbb{R}$ Isometry Group for a class of regular Cauchy data. In our first paper \cite{chen}, we reduce the Einstein equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof for the global regularity for the 2+1 dimensional system. Firstly, we show the non-concentration of the energy near the first possible singularity. Then, we prove that the global regularity holds for initial data with small energy.
Gerardo Sosa - One of the best experts on this subject based on the ideXlab platform.
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the Isometry Group of an rcd space is lie
Potential Analysis, 2018Co-Authors: Gerardo SosaAbstract:We give necessary and sufficient conditions that show that both the Group of isometries and the Group of measure-preserving isometries are Lie Groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by R C D ∗-spaces and, under extra assumptions, by C D-spaces, C D ∗ P-spaces. However, we show that the M C C P-condition by itself is not enough to guarantee a smooth behavior of these automorphism Groups.
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the Isometry Group of an mathsf rcd space is lie
arXiv: Differential Geometry, 2016Co-Authors: Gerardo SosaAbstract:We give necessary and sufficient conditions that show that both the Group of isometries and the Group of measure-preserving isometries are Lie Groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD*-spaces and, under extra assumptions, by CD-spaces, CD*-spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism Groups. More generally we show that spaces with good optimal transport properties meet as well the hypotheses.
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the Isometry Group of an mathsf rcd star space is lie
arXiv: Differential Geometry, 2016Co-Authors: Gerardo SosaAbstract:We give sufficient conditions to show that both the Group of isometries and the Group of measure-preserving isometries of a metric measure space are Lie Groups. In addition, we study in which cases spaces satisfying a generalized lower Ricci curvature bound satisfy these assumptions. The conditions are fulfilled in the class of $\mathsf{RCD}^\star$-spaces, however, we give an example of an $\mathsf{MCP}$-space for which these automorphism Groups are not Lie Groups.
