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).
Highlights
As is well known, Einstein spaces play a very important role in the general theory of relativity.The conformal mappings of these spaces has been studied since 1920 by Brinkmann [1], see [2,3].Brinkmann proved that this task is closely related to the existence of concircular vector fields.In 1944, Yano [4,5,6,7] introduced term a concircular vector field ξ, which satisfies ∇ξ = $ · Id, where ∇ is affine connection
A lot of work has been devoted to special mappings of Einstein spaces, such as [2,3,8,9,14,15,16,17,18,19,20,21,22,23,24]
Since by assumptions the function ψ is defined on the whole R and is equal to zero at no point, we have ψ = const along complete light-like geodesics
Summary
Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a domain V of n-dimensional manifold M. For the function ψ, the metric ψ−2 g is Einstein, ψ is a constant. For n = 2, it is trivial, and for n = 3 any Einstein space has the constant curvature.
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