TAILIEUCHUNG - On a class of Para-Sakakian manifolds

In the paper, the authors studied conformally symmetric Para-Sasakian manifolds and they proved that an n-dimensional conformally symmetric Para-Sasakian manifold is conformally flat and SP-Sasakian. | Turk J Math 29 (2005) , 249 – 257. ¨ ITAK ˙ c TUB On A Class of Para-Sakakian Manifolds ¨ ur Cihan Ozg¨ Abstract In this study, we investigate Weyl-pseudosymmetric Para-Sasakian manifolds and Para-Sasakian manifolds satisfying the condition C · S = 0. Key Words: Para-Sasakian manifold, Weyl-pseudosymmetric manifold. 1. Introduction Let (M, g) be an n-dimensional, n ≥ 3, differentiable manifold of class C ∞ . We denote by ∇ its Levi-Civita connection. We define endomorphisms R(X, Y ) and X ∧ Y by R(X, Y )Z = [∇X , ∇Y ]Z − ∇[X,Y ] Z, (1) (X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y, (2) respectively, where X, Y, Z ∈ χ(M ), χ(M ) being the Lie algebra of vector fields on M . The Riemannian Christoffel curvature tensor R is defined by R(X, Y, Z, W ) = g(R(X, Y )Z, W ), W ∈ χ(M ). Let S and κ denote the Ricci tensor and the scalar curvature of M , respectively. The Ricci operator S and the (0,2)-tensor S 2 are defined by g(SX, Y ) = S(X, Y ), (3) S 2 (X, Y ) = S(SX, Y ). (4) and 2000 Mathematics Subject Classification: 53B20, 53C15, 53C25 249 ¨ ¨ OZG UR The Weyl conformal curvature operator C is defined by C(X, Y ) = R(X, Y ) − κ 1 (X ∧ SY + SX ∧ Y − X ∧ Y ), n−2 n−1 (5) and the Weyl conformal curvature tensor C is defined by C(X, Y, Z, W ) = g(C(X, Y )Z, W ). If C = 0, n ≥ 4, then M is called conformally flat. For a (0, k)-tensor field T , k ≥ 1, on (M, g) we define the tensors R · T and Q(g, T ) by (R(X, Y ) · T )(X1 ,.,Xk ) = −T (R(X, Y )X1 , X2 ,.,Xk ) (X1 , ., Xk−1, R(X, Y )Xk ), Q(g, T )(X1 ,.,Xk ; X, Y ) = (6) -T ((X ∧ Y )X1 , X2 ,.,Xk ) (X1 ,.,Xk−1 , (X ∧ Y )Xk ), (7) respectively [8]. If the tensors R · C and Q(g, C) are linearly dependent then M is called Weylpseudosymmetric. This is equivalent to R · C = LC Q(g, C), (8) holding on the set UC = {x ∈ M

• Toán học
• Para Sasakian manifold
• Weyl pseudosymmetric manifold
• Conformally symmetric
• Conformally flat
• Conformal curvature
• Degenerate metric
• Screen conformal lightlike hypersurface
• Parallel lightlike hypersurface
• Semi symmetric lightlike hypersurface
• Minkowski spacetime