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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.

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)