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Homogeneity of Lorentzian three-manifolds with recurrent curvature. (English) Zbl 1307.53057
The authors study the homogeneity of Lorentzian three-manifolds with recurrent curvature. The curvature tensor $$R$$ of a pseudo-Riemannian manifold is called recurrent if $$\nabla R=\omega\otimes R$$, where $$\omega$$ is a $$1$$-form. A pseudo-Riemannian manifold is called $$k$$-curvature homogeneous if for any two points there exists a linear isometry between the corresponding tangent spaces which preserves the curvature tensor and its derivatives upt to order $$k$$. A Lorentzian manifold is called a Walker manifold if it admits a parallel null vector field. The authors obtain a complete description of $$k$$-curvature homogeneous three-dimensional Walker metrics with $$k\leq 2$$. This leads to a complete description of locally homogeneous three-dimensional Walker metrics as well as a complete description of all locally homogeneous such metrics with recurrent curvature. Some results related to steady gradient Ricci and Cotton solitons are also deduced.

##### MSC:
 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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