JOURNAL Series A - Pure and Applied Mathematics

c○Journal of Technical University at PlovdivFundamental Sciences and Applications, Vol. 11, 2005-2006Series A-Pure and Applied MathematicsBulgaria, ISSN 1310-8271Some properties of one connection onspaces with an almost product structureIva DokuzovaAbstractIn a space M with a metric g and an almost product structure J weintroduce an affine connection by using the Levi-Civita connection ∇ of g.We get some properties of the obtained transformation.Keywords: semi-Riemannian geometry, curvature tensor, geodesics, Einsteinfield equations, Schwarzschild solution.1 IntroductionWe consider a space M(dim M = n) with a metric g and an almost product structureJ, which preserves the scalar product. SoJ 2 = id (J ≠ id), g(Jx, Jy) = g(x, y), x, y ∈ χM. (1)Let ∇ be the Levi-Civita connection of g and R be the curvature tensor field of ∇.If ∇ satisfies∇J = 0, (2)then M is a locally decomposable Riemannian space [2], [3], i.e. M = M 1 × M 2 ,where M 1 , M 2 are both Riemannian spaces. We note that trJ = dim M 1 − dim M 2 ,n = dim M 1 + dim M 2 .0 2000 Mathematics Subject Classification: 53C15, 53B05,53B30.0 Key words and phrases: semi-Riemannian geometry, curvature tensor, geodesics, Einstein fieldequations, Schwarzschild solution.0 Received November 10, 2006