On weak symmetries of Einstein and Sasakian manifolds.(English)Zbl 0849.53038

In a previous paper, the authors called a Riemannian manifold $$M$$ weakly symmetric if $$\nabla R$$ can be expressed by the curvature tensor $$R$$ in a certain way involving a number of 1-forms [Colloq. Math. Soc. János Bolyai 56, 663-670 (1992; Zbl 0791.53021)]. If, in addition, $$M$$ is either Einstein or Sasakian, then a linear relation between these 1-forms is deduced. Analogous results hold under the same weak symmetry assumption on $$\nabla\text{ Ric}$$.

MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53B20 Local Riemannian geometry

Zbl 0791.53021