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On pseudosymmetric para-Kählerian manifolds. (English) Zbl 1076.53034
A para-Kählerian manifold is a triple \((M,J,g)\), where \(M\) is a differentiable manifold of dimension \(n=2m\), \(J\) a \((1,1)\)-tensor field and \(g\) a pseudo-Riemannian metric on \(M\) satisfying the conditions \( J^2=I\), \(g(JX,JY)=-g(X,Y)\), \(\nabla J=0\). Let \(R,S\) be the curvature tensor and the Ricci tensor of \(g\) respectively. For a \((0,k)\)-tensor \((k\geq 1)\) field \(T\) one defines the \((0,k+2)\)-tensor field \(R\cdot T\) by the condition \[ (R\cdot T)(U,V,X_1,\dots,X_k)=-\sum_{s=1}^{k} T(X_1,\dots,R(U,V)X_s,\dots,X_k) \] and the \((0,k+2)\)-tensor field \(Q(g,T)\) by \[ Q(g,T)(U,V,X_1,\dots,X_k)=-\sum_{s=1}^{k}T(X_1,\dots,(U\wedge V)X_s,\dots,X_k). \] The manifold is called semisymmetric if \(R\cdot R=0\), Ricci-semysimmetric if \(R\cdot S=0\), pseudosymmetric (respectively, Ricci pseudosymmetric) if there exists a function \(L: M \rightarrow \mathbb R\) such that \(R\cdot R=LQ(g,R)\) (respectively, \(R\cdot S=LQ(g,S)\)). Analogous definitions involving (semi or pseudo)symmetry with respect to the Bochner or the paraholomorphic projective curvature tensor can be considered.
The author proves several results where pseudosymmetry imply semi-symmetry. As an example: A pseudosymmetric para-Kählerian manifold \((M,J,g)\) of dimension \(>4\) is semisymmetric. Moreover, new examples of para-Kählerian manifolds which are Ricci-semisymmetric (in dimensions \(\geq 6\)) or pseudosymmetric (in dimension 4) or Bochner-pseudosymmetric (in dimension 4) are given.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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