A Walker n-manifold is a semi-Riemannian manifold, which admits a field of parallel null r-planes, with . In the present paper we study curvature properties of a Walker 4-manifold (M, g) which admits a field of parallel null 2-planes. The metric g is necessarily of neutral signature (+ + − −). Such a Walker 4-manifold is the lowest dimensional example not of Lorentz type. There are three functions of coordinates which define a Walker metric. Some recent work shows that a Walker 4-manifold of restricted type whose metric is characterized by two functions exhibits a large variety of symplectic structures, Hermitian structures, Kähler structures, etc. For such a restricted Walker 4-manifold, we shall study mainly curvature properties, e.g., conditions for a Walker metric to be Einstein, Osserman, or locally conformally flat, etc. One of our main results is the exact solutions to the Einstein equations for a restricted Walker 4-manifold.
We consider four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds. According to forms (Segre types) of the Ricci operator, we provide a full classification of four-dimensional pseudo-Riemannian conformally flat homogeneous Ricci solitons.2000 Mathematics Subject Classification. 53C50, 53C15, 53C25.
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