zbMATH — the first resource for mathematics

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.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C35 Differential geometry of symmetric spaces