Examples of signature (2,2) manifolds with commuting curvature operators

Miguel Brozos-Vazquez, Eduardo Garcia-Rio, Peter B. Gilkey, and Ramon Vazquez-Lorenzo

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Submission date: 17. Aug. 2007
Pages: 12
published in: Journal of physics / A, 40 (2007) 43, p. 13149-13159 
DOI number (of the published article): 10.1088/1751-8113/40/43/021
Bibtex
MSC-Numbers: 53C20
Keywords and phrases: anti-self-dual, self-dual, conformal Osserman, Einstein, curvature--Jacobi--Ricci commuting, Osserman, Ricci operator
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Abstract:
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure, these properties are related to the Ricci tensor of the affine structure.