Abstract.
We call a pseudo-Riemannian 4-manifold, which admits a field of parallel null 2-planes, a Walker 4-manifold. A pseudo-Riemannian metric of a Walker 4-manifold is necessarily of neutral signature, and it admits an orthogonal almost complex structure. We show that such a Walker 4-manifold can carry various structures with respect to a certain kind of almost complex structure, e.g., symplectic structures, Kähler structures, Hermitian structures, according as the properties of certain functions which define the canonical form of the metric. The combination of these structures are also analyzed.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Haze passed away on 14 March 2006.
Rights and permissions
About this article
Cite this article
García-Río, E., Haze, S., Katayama, N. et al. Symplectic, Hermitian and Kähler Structures on Walker 4-Manifolds. J. geom. 90, 56–65 (2008). https://doi.org/10.1007/s00022-008-1999-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-008-1999-y