$\varphi (\mbox {\textbf{Ric}})$-vector fields in Riemannian spaces

Irena Hinterleitner and Volodymyr A. Kiosak

Faculty of Mechanical Engineering, Brno University of Technology Technick√° 2896/2, 616 69 Brno, Czech Republic
Friedrich-Schiller-Universität Jena, Mathematisches Institut Ernst-Abbe-Platz 2, 07743 Jena, Germany


Abstract: 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.

AMSclassification: primary 53B05; secondary 53B30.

Keywords: special vector field, pseudo-Riemannian spaces, Riemannian spaces, symmetric spaces, Kasner metric.