Curvature properties of four-dimensional Walker metrics

Details Curvature properties of four-dimensional Walker metrics Curvature properties of four-dimensional Walker metrics

Feb 2005

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005cqgra..22..559c&link_type=abstract

Classical and Quantum Gravity, Volume 22, Issue 3, pp. 559-577 (2005).

Physics

9

Scientific paper

A Walker n-manifold is a semi-Riemannian manifold, which admits a field of parallel null r-planes, with r \leq \frac{n}{2} . 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.

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