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Generalized recurrent and concircular recurrent manifolds. (English) Zbl 0981.53030
Summary: The concepts of generalized recurrent manifold $$GK_n$$ and generalized concircular recurrent manifold $$G(ZK_n)$$ are considered to prove that a manifold $$(M_n,g)$$ which is either $$GK_n$$ or $$G(ZK_n)$$ is a concircular recurrent manifold $$ZK_n$$. By using a result due to P. Desai and K. Amur [Tensor, New Ser. 29, 98-102 (1975; Zbl 0309.53022)], it is claimed that such a manifold $$(M_n,g)$$ is either a recurrent space $$K_n$$ or an Einstein space. Further it is noticed that the condition $$A(lx)+(n-1) B(x)=0$$ used by U. C. De and D. Kamilya [Bull. Calcutta Math. Soc. 86, 69-72 (1994; Zbl 0815.53025)] in proving the result is not required.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
##### Keywords:
Einstein space; concircular recurrent manifold