Ric-vector fields in Riemannian spaces. (English) Zbl 1212.53018

Hinterleitner, Irena; Kiosak, Volodymyr A.

Arch. Math., Brno 44, No. 5, 385-390 (2008).

Summary: We study vector fields in Riemannian spaces which satisfy \(\nabla \mathbf {\varphi }=\mu\), \({\mathbf {Ric}}\), \(\mu =\text{const.}\) We investigate the properties of these fields and conditions for their coexistence with concircular vector fields. It is shown that in Riemannian spaces, noncollinear concircular and \(\mathbf {\varphi }(\mathbf {Ric})\)-vector fields cannot exist simultaneously. It is found that Riemannian spaces with \(\mathbf {\varphi }(\mathbf {Ric})\)-vector fields of constant length have constant scalar curvature. Conditions for the existence of \(\mathbf {\varphi }(\mathbf {Ric})\)-vector fields in symmetric spaces are given.

MSC:

53B05	Linear and affine connections
53B30	Local differential geometry of Lorentz metrics, indefinite metrics

Keywords:

special vector field; pseudo-Riemannian space; Riemannian space; symmetric space; Kasner metric