Properties of the Weyl conformally curvature of Kähler-Norden manifolds (Q2733955)

Let \(M(J,g)\) be an \(n=2m\)-dimensional Kählerian manifold endowed with a Norden metric. This means, \(J\) is a complex structure on \(M\) and \(g\) is a pseudo-Riemannian metric of signature \((m,m)\) for which \(J\) is an antiisometry, i.e. \(g(JX,JY)=-g(X,Y)\), and \(\nabla J=0\) [cf. e.g. \textit{G. T. Ganchev} and \textit{A. V. Borisov}, C. R. Acad. Bulg. Sci. 39, No. 5, 31-34 (1986; Zbl 0608.53031)]. NEWLINENEWLINENEWLINEIt is proved that \(M\) is conformally flat if and only if it is holomorphically projectively flat and its scalar curvature vanishes. The so-called \(\ast\)-scalar curvature of such a manifold is constant and \(M\) is locally symmetric. If \(M\) is of recurrent conformal curvature, then it is locally symmetric or \(n=4\) and the manifold is holomorphically projectively flat. Moreover, both the pseudosymmetry (in the sense of \textit{R. Deszcz} [Bull. Soc. Math. Belg., Sér. A 44, 1-34 (1992; Zbl 0808.53012)]) as well as the Weyl-pseudosymmetry and the holomorphically projective-pseudosymmetry of \(M\) always reduce to semisymmetry and the Ricci-pseudosymmetry of \(M\) reduces to the Ricci-semisymmetry. An example of a Ricci-semisymmetric and non-semisymmetric Kähler-Norden structure is stated. Examples of semisymmetric, especially locally symmetric, Kähler-Norden structures are given by the author in a previous paper [An. Ştiint. Univ. Al. I. Cuza Iaşi, Ser. Ia Mat (in print)].NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].