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A note on conformally quasi-recurrent manifolds. (English) Zbl 0757.53008

A (pseudo-)Riemannian manifold \((M,g)\) is said to be conformally recurrent if its conformal curvature tensor \(C\) satisfies the condition \(\nabla_ hC_{ijkl}=a_ hC_{ijkl}\), and \((M,g)\) is called conformally quasi-recurrent if \[ \nabla_ hC_{ijkl}=2a_ hC_{ijkl}+a_ iC_{hjkl}+a_ jC_{ihkl}+a_ kC_{ijhl}+a_ lC_{ijkh} . \] In both the above conditions \(a\) is a differential 1-form on \(M\). The author finds conditions for a conformally changed conformally quasi-recurrent metric to be conformally quasi-recurrent as well as conformally symmetric \((\nabla C=0)\). Moreover, based on a classification by A. Derdziński [Colloq. Math. 42, 59-81 (1979; Zbl 0439.53034), ibid. 44, 77-95 (1981; Zbl 0491.53013) and 44, 249-262 (1981; Zbl 0491.53014)] she describes a procedure to obtain a conformally quasi- recurrent metric by a conformal deformation of a conformally symmetric metric. Finally, properties of totally umbilical hypersurfaces in conformally quasi-recurrent manifolds are studied.

MSC:

53B20 Local Riemannian geometry
53B25 Local submanifolds
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