In the present paper we prove non-existence theorems for conformal mappings of compact (pseudo-)Riemannian manifolds onto Einstein manifolds without boundary. We obtained certain conditions for which these mappings are only trivial.
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).
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