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special vector field; pseudo-Riemannian spaces; Riemannian spaces; symmetric spaces; Kasner metric
In this paper we study vector fields in Riemannian spaces, which satisfy $\nabla \varphi =\mu $, ${\textbf{Ric}}$, $\mu =\mbox {const.}$ We investigate the properties of these fields and the conditions of their coexistence with concircular vector fields. It is shown that in Riemannian spaces, noncollinear concircular and $\varphi (\mbox {\textbf{Ric}})$-vector fields cannot exist simultaneously. It was found that Riemannian spaces with $\varphi (\mbox {\textbf{Ric}})$-vector fields of constant length have constant scalar curvature. The conditions for the existence of $\varphi (\mbox {\textbf{Ric}})$-vector fields in symmetric spaces are given.
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