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Deformations of three-dimensional metrics. (English) Zbl 1317.83063
Summary: We examine three-dimensional metric deformations based on a tetrad transformation through the action the matrices of scalar field. We describe by this approach to deformation the results obtained by B. Coll et al. [Gen. Relativ. Gravitation 34, No. 2, 269–282 (2002; Zbl 1008.53039)], where it is stated that any three-dimensional metric was locally obtained as a deformation of a constant curvature metric parameterized by a 2-form. To this aim, we construct the corresponding deforming matrices and provide their classification according to the properties of the scalar $$\sigma$$ and of the vector $$\mathbf {s}$$ used in Coll et al. [loc. cit.] to deform the initial metric. The resulting causal structure of the deformed geometries is examined, too. Finally, we apply our results to a spherically symmetric three geometry and to a space sector of Kerr metric.

##### MSC:
 83C80 Analogues of general relativity in lower dimensions 53Z05 Applications of differential geometry to physics 83C57 Black holes
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