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Bochner flat Kählerian manifolds with a certain condition on the Ricci tensor. (English) Zbl 0629.53059
Let M be a Bochner flat Kähler manifold. For the Ricci tensor $$\rho$$ of M, the condition $$(*)\quad R\cdot \sigma =fR^ 1\cdot \rho$$ is studied, R being the curvature operator, f a function and $R^ 1_{XY}=X\wedge Y+(JX)\wedge (JY)-2g(JX,Y)J$ for X,Y$$\in {\mathcal X}(M)$$. Among other things, it is proved that the condition (*) is fulfilled on M iff $$\rho$$ has at most two distinct eigenvalues. It is also shown that if M is compact, dim $$M\geq 6$$ and M satisfies the condition (*) with $$f\leq 0$$, then M is locally symmetric. All currently known Bochner flat Kählerian manifolds satisfy the condition.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53B35 Local differential geometry of Hermitian and Kählerian structures