There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics

Abstract

In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ −2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel).

Conformal Equitorsion and Concircular Transformations in a Generalized Riemannian Space

“…De and Mandal in [3] studied concircular curvature tensors as important tensors from the differential geometric point of view. In [4][5][6][7][8][9][10][11], Mikeš et al have studied special kinds ot transformations in Riemannian space.…”

Geodesic Vector Fields on a Riemannian Manifold

“…Therefore, we could expect that geodesic vector fields also have the scope of applications in general relativity. For example, global questions about the existence of these vector fields were studied in [5][6][7][8][9][10]. 4.…”

“…Similarly, special vector fields such as unit geodesic vector fields, Killing vector fields, concircular vector fields, conformal vector fields are used in studying geometry as well as topology of a Riemannian manifold (cf. [1][2][3][4][6][7][8][9][10][11][16][17][18][19][20][21][22][23][24][25][26][27][28][29]).…”