This paper is concerned with almost para-complex structures on Walker 4-manifolds. For these structures, we study some problems of Kahler manifolds. We also give an example of a flat almost para-complex manifold, which consists of a nonintegrable almost para-complex structure on Walker 4-manifolds. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 1071 – 1080 ¨ ITAK ˙ c TUB ⃝ doi: On 4-dimensional almost para-complex pure-Walker manifolds ˙ ¸ CAN1,∗, Hilmi SARSILMAZ1 , Sibel TURANLI2 Murat IS Department of Mathematics, Faculty of Sciences, Atat¨ urk University, Erzurum, Turkey 2 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, Turkey 1 Received: • Accepted: • Published Online: • Printed: Abstract: This paper is concerned with almost para-complex structures on Walker 4-manifolds. For these structures, we study some problems of K¨ ahler manifolds. We also give an example of a flat almost para-complex manifold, which consists of a nonintegrable almost para-complex structure on Walker 4-manifolds. Key words: Almost para-complex structure, pure metric, neutral metric, Walker metric, K¨ ahler structure 1. Introduction Let M2n be a semi-Riemannian smooth manifold with the metric g , which is necessarily of neutral signature (n, n), and let ℑrs (M2n ) be the tensor field of M2n , . the field of all tensors of type (r, s) in M2n . An almost para-complex structure on M2n is an affinor field φ on M2n : φ2 = I , and the 2 eigenbundles T M2n and T − M2n corresponding to the two eigenvalues +1 and –1 have the same rank. The pair (M2n , φ) is called an almost para-complex manifold. + Let (M2n , φ) be an almost para-complex manifold with almost para-complex structure φ. If the Nijenhuis tensor of such a affinor field φ defined by Nφ (X, Y ) = [φX, φY ] − φ[φX, Y ] − φ[X, φY ] + [X, Y ] is equivalent to the vanish, for any vector fields X , Y on M2n , then the almost para-complex structure φ is integrable and it is said to be a para-complex structure. . Pure metrics A pure metric with respect to the almost para-complex structure is a semi-Riemannian metric g such that g(φX, φY ) = g(X, Y ) (1) for .
