Left-invariant lorentz metrics on lie groups pdf

Lorentzian left invariant metrics on three dimensional unimodular lie groups through this section, for any g among th e. However, the study of left invariant einstein pseudoriemannian metrics on lie groups is at beginning. O1,2 is lorentz group consisting of linear transformations the quadratic from t2. However, relying on the choice of left invariant lorentz metrics, the sign of the.

Curvatures of left invariant metrics on lie groups. Lie groups, lie algebras, flat lorentzian metric, killing vector field. Lorentz manifolds, manifolds with indefinite metrics. However, groups with such metrics are always noncompact as a consequence of bochners vanishing theorem. In this paper, we consider lorentz metrics on 3dimensional lie groups. When the manifold is a lie group and the metric is left invariant the curvature. Curvature of left invariant riemannian metrics on lie groups. Curvatures of left invariant metrics on lie groups john milnor institute for advanced study, princeton, new jersey 08540 this article outlines what is known to the author about the riemannian geometry of a lie group which has been provided with a riemannian metric invariant under left translation. On the isometry groups of invariant lorentzian metrics on the.

We investigate their geometry, especially holonomy groups and decomposability of these metrics. Harmonic tensors on threedimensional lie groups with left. Pdf on lie groups with left invariant semiriemannian metric. Advances in mathematics 21,293329 1976 curvatures of left invariant metrics on lie groups john milnor institute for advanced study, princeton, new jersey 08540 this article outlines what is known to the author about the riemannian geometry of a lie group which has been provided with a riemannian metric invariant under left translation. Curvatures of left invariant metrics on lie groups john milnor. In 2, 1, flat lorentzian left invariant metrics on lie groups has been studied, in 5 flat.

Lorentz geometry of 4dimensional nilpotent lie groups. We find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. Contracting the schoutenweyl tensor in an arbitrary direction, we introduce an antisymmetric 2tensor and study the structure of threedimensional lie groups and algebras with leftinvariant riemann metric in which this tensor is harmonic. Lorentzian flat lie groups admitting a timelike leftinvariant killing. Leftinvariant lorentzian flat metrics on lie groups. A criterion for global hyperbolicity of leftinvariant. Ricciflat leftinvariant lorentzian metrics 4pt on 2step nilpotent lie. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. All known examples of lie groups admitting leftinvariant riemannian einstein metrics with negative einstein constants lie group.

Heisenberg lie group of dimension three h3r and the biinvariant metrics on the. We study threedimensional lie groups with leftinvariant lorentz metric and almost harmonic with zero curl and divergence schoutenweyl tensor. We now define a lorentzian metric1 on m giving its components relative to the basis e1. Leftinvariant lorentz metrics on lie groups project euclid. In this section, we will give a criterion for a leftinvariant pseudoriemannian metric on a lie group to be globally hyperbolic. Lie groups which can carry a flat leftinvariant lorentzian metric i. On lie groups with left invariant semiriemannian metric r. Heisenberg group h3 has three leftinvariant lorentzian metrics. Pdf leftinvariant lorentzian metrics on 3dimensional. Leftinvariant lorentz metrics on lie groups core reader.