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On quasi-conformally flat weakly Ricci symmetric manifolds. (English) Zbl 1164.53011

The authors obtain certain properties of (pseudo-)Riemannian manifolds which are weakly Ricci symmetric and simultaneously quasi-conformally flat. The reader must be careful because the formulations of the theorems are not complete. For instance, Theorem 1 states that the Ricci tensor is of rank 1. In this theorem, it is in fact additionally assumed that \(\delta\neq0\), which eliminates a large subclass of the considered manifolds. Moreover, the examples are not well described. For instance, the metric in Example 1 is Ricci recurrent, which contradicts Theorem 11, where the non-Ricci recurrence of this metric is asserted.

MSC:

53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B35 Local differential geometry of Hermitian and Kählerian structures
53B05 Linear and affine connections
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