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On generalized pseudo symmetric manifolds. (English) Zbl 0889.53015
A Riemannian space $$V^n$$ is locally symmetric if $$\nabla_XR=0$$, and of recurrent curvature if $$(*)$$ $$\nabla_XR=a(X)R$$ for some 1-form $$a$$. This notion was weakened by M. C. Chaki [Publ. Math. 45, 305-312 (1994; Zbl 0827.53032)] and L. Tamássy and T. Q. Binh [Colloq. Math. Soc. János Bolyai 56, 663-670 (1992; Zbl 0791.53021)] by replacing $$(*)$$ with $$(\nabla_XR)(Y,V,W,Z)= \sigma(X,Y,V,W,Z)$$ $$\{\overset\sigma a(X)R(Y,V,W,Z)\}$$ ($$\sigma$$ means the cyclic summation). Such a $$V^n$$ is called a generalized pseudosymmetric space $$G(PS)_m$$. Now it is proved that some of the 1-forms $$\overset\sigma a$$ of a $$G(PS)_n$$ necessarily coincide. Conformally flat $$G(PS)_n$$ with constant scalar curvature are also investigated.

##### MSC:
 53B20 Local Riemannian geometry