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On weakly quasi-conformally symmetric manifolds. (English) Zbl 1091.53017
Let $$(M,g)$$ be a Riemannian manifold. The quasi-conformal tensor $$W$$ is a linear combination with constant coefficients of the conformal Weyl tensor and the quasi-circular curvature tensor. The authors consider a class of Riemannian manifolds such that the covariant derivative $$\nabla W$$ satisfies the identity $(\nabla_{X_1} W)(X_2,X_3,X_4,X_5) = \operatorname{cycl} \alpha_i(X_i)W(X_j, X_k,X_l,X_m)$ where $$X_1,X_2,X_3,X_4,X_5 \in TM$$ , $$\alpha_i$$, $$i = 1,2,3,4,5$$ are 1-forms and cycl stands for a sum of cyclic permutations of $$1,2,3,4,5$$. They give some examples of such manifolds, study their properties and derive conditions when it is a weakly projectively symmetric manifold in the sense of L. Tamassy and T.Q. Binh.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C35 Differential geometry of symmetric spaces
##### Keywords:
Quasi-conformally symmetric manifolds