On Compact Holomorphically Pseudosymmetric Kählerian Manifolds

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold. AMS Mathematics Subject Classification (2000) 53C55, 53C25 1. Holomorphic pseudosymmetries Let M be a 2n-dimensional Kählerian manifold with (J, g) as its Kählerian structure. Thus, J is a (1, 1)-tensor field (an almost complex structure) and g a Riemannian metric on M such that J = −I, g(J ·, J ··) = g(·, ··) and ∇J = 0, ∇ being the Levi-Civita connection of g. Let X(M) be the Lie algebra of smooth vector fields on M . For U, V ∈ X(M), let R(U, V ) = [∇U ,∇V ]−∇[U,V ] = ∇ 2 UV −∇ 2 V U be the usual curvature operator, and consider additional curvature type operator R(U, V ) defined by assuming that (1) R(U, V )X = g(V,X)U − g(U,X)V + g(JV,X)JU − g(JU,X)JV − 2g(JU, V )JX for any X ∈ X (M). The operators R(U, V ) and R(U, V ) will be treated as derivations of the tensor algebra on M in the usual sense. For instance, if T is an (0, k)-tensor field, then R(U, V )T , R(U, V )T are the (0, k)-tensor fields such that (R(U, V )T )(X1, . . . , Xk) = − ∑ s T (X1, . . . , Xs−1,R(U, V )Xs, Xs+1, . . . , Xk), (R(U, V )T )(X1, . . . , Xk) = − ∑ s T (X1, . . . , Xs−1,R (U, V )Xs, Xs+1, . . . , Xk). For an (0, k)-tensor field T , define (0, k + 2)-tensor fields R · T , R · T by (R · T )(U, V,X1, . . . , Xk) = (R(U, V ) · T )(X1, . . . , Xk) (R · T )(U, V,X1, . . . , Xk) = (R (U, V ) · T )(X1, . . . , Xk). Let us call an (0, k)-tensor field T on M to be • semisymmetric if R · T = 0; • holomorphically pseudosymmetric if there exists a function f (called the structure function) on M such that R · T = fR · T . A Kählerian manifold will be called • semisymmetric (resp., Ricci-semisymmetric) if its Riemann (resp., Ricci) curvature tensor is semisymmetric; • holomorphically pseudosymmetric (resp., Ricci-pseudosymmetric) if its Riemann (resp., Ricci) curvature tensor is holomorphically pseudosymmetric.