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

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