Article

Full entry | PDF   (0.4 MB)
Keywords:
special vector field; pseudo-Riemannian spaces; Riemannian spaces; symmetric spaces; Kasner metric
Summary:
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.
References:
[1] Brinkmann, H. W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94 (1) (1925), 119–145. DOI 10.1007/BF01208647 | MR 1512246
[2] Hall, G.: Some remarks on the space-time of Newton and Einstein. Fund. Theories Phys. 153 (2007), 13–29. MR 2368238 | Zbl 1151.83001
[3] Kiosak, V. A.: Equidistant Riemannian spaces. Geometry of generalized spaces. Penz. Gos. Ped. Inst., Penza (1992), 37–41. MR 1265502
[4] Landau, L. D., Lifschitz, E. M.: Lehrbuch der theoretischen Physik, II, Klassische Feldtheorie. Akademie Verlag, Berlin, 1973. MR 0431969
[5] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. 78 (3) (1996), 311–333. DOI 10.1007/BF02365193 | MR 1384327
[6] Mikeš, J., Hinterleitner, I., Kiosak, V. A.: On the theory of geodesic mappings of Einstein spaces and their generalizations. AIP Conf. Proc., 2006, pp. 428–435.
[7] Mikeš, J., Rachůnek, L.: On tensor fields semiconjugated with torse-forming vector fields. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 44 (2005), 151–160. MR 2218574 | Zbl 1092.53016
[8] Mikeš, J., Škodová, M.: Concircular vector fields on compact spaces. Publ. de la RSME 11 (2007), 302–307.
[9] Shandra, I. G.: Concircular vector fields on semi-riemannian spaces. J. Math. Sci. 31 (2003), 53–68. MR 2464554
[10] Yano, K.: Concircular Geometry. I-IV. Proc. Imp. Acad., Tokyo, 1940. Zbl 0025.08504

Partner of