ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2007, Vol. 42, No. 5, pp. 270–277. c  Allerton Press, Inc., 2007. Original Russian Text c  S. Azimpour, M. Chaichi, M. Toomanian, 2007, published in Izvestiya NAN Armenii. Matematika, 2007, No. 5, pp. 42–52. FUNCTIONAL ANALYSIS A Note on 4-dimensional Locally Conformally Flat Walker Manifolds S. Azimpour 1* , M. Chaichi 2** , and M. Toomanian 3*** 1 University of Tabriz, Iran 2 University of Payame Noor of Tabriz, Iran 3 University of Tabriz, Iran (Azad University, Karaj) Received May 10, 2007 Abstract—A 4-dimensional Walker metric on a semi Riemanian manifold M , for the canonical metric with c =0, have been investigated by M. Chaichi, E. Garc ´ ıa–R ´ ıo and Y. Matsushita. The paper generalizes these notions to the case of constant c =0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found. MSC2000 numbers : 53C55 DOI: 10.3103/S1068362307050044 Key words: Indefinite metrics; curvature tensor; locally conformally flat; Walker manifold. 1. INTRODUCTION A Walker n-manifold, admits a field of parallel null r-planes, with r ≤ n 2 . The canonical forms of the metrics have been investigated by Walker [4]. The canonical form of this metric contain three functions a(x,y,z,t), b(x,y,z,t) and c(x,y,z,t). Our interest is in 4-dimensional Walker manifolds with parallel null 2-planes. In [2], Einstein, Osserman and locally conformally flat Walker manifolds were investigated in the restricted form of metric when c(x,y,z,t)=0. Following [2], in this paper we focus on the case c(x,y,z,t)=c and investigate locally conformally flat 4-dimensional Walker manifolds admitting parallel null 2-plane. Definition 2. A Walker manifold is a triple (M,g,D) consisting of an n-dimensional manifold M , an indefinite metric g and an r-dimensional parallel distribution D. If dim M =4 and dim D =2, g has signature (−− ++) and in suitable coordinates, g can be given by g(x,y,z,t)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 00 1 0 00 0 1 10 a(x,y,z,t) c(x,y,z,t) 01 c(x,y,z,t) b(x,y,z,t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (1.1) for some functions a(x,y,z,t), b(x,y,z,t), c(x,y,z,t) and D =  ∂ ∂x , ∂ ∂y  (ref. [4], [5]). The special case where c(x,y,z,t) ≡ 0 was studied in [2], where the components of Levi–Civita connection, and the curvature tensor has been calculated and the conditions that creates locally conformally flat are given. We are interested in the case of the Walker metric g as in (1.1) with constant c. * E-mail: sohrab_azimpour@yahoo.com ** E-mail: chaichi@tabrizu.ac.ir *** E-mail: Toomanian@tabrizu.ac.ir 270