Symplectic, Hermitian and Kähler Structures on Walker 4-Manifolds

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. Mathematics Subject Classification (2000). 53C50, 53D05, 32Q20.

A Survey on Geometric Dynamics of 4-Walker Manifold

“…is named a complex Hamiltonian mechanical system on Walker manifold M 4 . We obtain the following corollary considering the equations found in (21) using Remark (p. 387) in [14] and Proposition 4 in [8] and Corollary 4 in [15].…”

Almost Kähler eight-dimensional Walker manifold

“…In [6], the authors studied a particular almost complex structure J. For this, they explicitly solved the PDEs for the fundamental 2-form to be closed (the almost Kähler condition), and they gave the integrability condition of J, which looks similar to the almost Kähler conditions.…”

“…Among these, the significant Walker manifolds are the examples of the non-symmetric and non-homogeneous Osserman manifolds [2]. It was shown in [6,9,10] that the Walker 4-manifolds of neutral signature admit a pair comprising of an almost complex structure and an opposite almost complex structure, and that Petean's non-flat indefinite Kähler-Einstein metric on a torus was obtained as an example of Walker 4-manifolds. Banyaga and Massamba in [1] derived a Walker metric when studying the non-existence of certain Einstein metrics on some symplectic manifolds.…”

Some properties of four-dimensional Walker manifolds

“…Among these, the significant Walker manifolds are examples of the non-symmetric and non-homogeneous Osserman manifolds [2,3]. Recently, it was shown [4,6,7] that the Walker 4-manifolds of neutral signature admit a pair comprising an almost complex structure and an opposite almost complex structure, and that Petean's nonflat indefinite Kahler-Einstein metric on a torus was obtained as an example of a Walker 4-manifold. Moreover, Banyaga and Massamba derived in [1] a Walker metric when studying the non-existence of certain Einstein metrics on some symplectic manifolds.…”