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Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist

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Abstract

We show that a germ of a real-analytic Lorentz metric on \({\mathbb R}^3\) which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously—meaning they act transitively on an open set admitting the origin in its closure, but not at the origin—and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneous action is semisimple, we provide a complete set of normal forms of the metric and the action.

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Authors and Affiliations

  1. Laboratoire J.-A. Dieudonné, UMR 7351, Université Côte d’Azur, Université Nice Sophia Antipolis, CNRS, Parc Valrose, 06108, Nice Cedex 2, France

    Sorin Dumitrescu

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Karin Melnick

Authors
  1. Sorin Dumitrescu
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  2. Karin Melnick
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Correspondence to Karin Melnick.

Additional information

The authors acknowledge support from U.S. National Science Foundation grants DMS-1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties (the GEAR Network).” Melnick was also supported during work on this project by a Centennial Fellowship from the American Mathematical Society and by NSF Grants DMS-1007136 and 1255462.

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Dumitrescu, S., Melnick, K. Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist. Geom Dedicata 179, 229–253 (2015). https://doi.org/10.1007/s10711-015-0078-4

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