There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics

Volodymyr Kiosak; Vladimir S. Matveev

doi:10.1016/j.crma.2009.06.017

Differential Geometry/Mathematical Physics

There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics
[Il n'existe pas de rescaling conformément Einstein d'une métrique d'Einstein pseudo-Riemannienne complète]

Présenté par : Étienne Ghys

Volodymyr Kiosak ; Vladimir S. Matveev

Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1067-1069.

Abstract (VO)
Résumé

Let g be an Einstein metric of indefinite signature such that the conformally-equivalent metric $ψ^{- 2} g$ is also Einstein. We show that if the metric g is light-line complete, then the conformal coefficient ψ is constant. If the manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš–Radulovich and Kühnel, but our proof is much simpler).

The proof is based on the investigation of the behavior of the function ψ along light-line geodesics: we show that for every light-line geodesic $γ (t)$ we have $ψ (γ (t)) = {const}_{1} \cdot t + {const}_{2}$ . Since the function ψ cannot vanish, the light-line completeness of the metric implies $ψ = {const}_{2}$ .

If the manifold is closed, the function ψ accepts its maximal value $ψ_{\max}$ at a certain point. Then, for every light-line geodesic γ through this point we have ${const}_{1} = 0$ implying $ψ = ψ_{\max}$ at every point of this geodesic. Repeating the argumentation, we obtain that for every light-line geodesic $γ_{1}$ intersecting γ we have $ψ = ψ_{\max}$ at every point of $γ_{1}$ as well and so on. Since every two points can be connected by a sequence of light-line geodesics, ψ is constant on the whole manifold.

Soit g une métrique pseudo-riemannienne non définie de type Einstein telle que la métrique conformément équivalente $ψ^{- 2} g$ soit aussi d'Einstein. Nous montrons que si la métrique g est lumière-complète, i.e. ses géodésiques isotropes sont complètes, alors le coefficient ψ est constant. Si la variété est fermée, l'hypothèse de complétude peut être omise (ce dernier résultat est dû à Mikeš–Radulovich et Kühnel, mais notre démonstration est plus courte).

La démonstration est basée sur l'étude du comportement de la fonction ψ le long des géodésiques de type lumière. Si $γ (t)$ est une telle géodésique, alors : $ψ (γ (t)) = {const}_{1} \cdot t + {const}_{2}$ . Comme la fonction ψ est non-nulle, la lumière-complétude implique $ψ = {const}_{2}$ .

Si la variété est fermée, la fonction ψ prend sa valeur maximale $ψ_{\max}$ en un certain point. Donc, pour toute géodésique de type lumière γ passant par ce point, on a ${const}_{1} = 0$ , ce qui implique que $ψ = ψ_{\max}$ en tout point de cette géodésique. En répétant cet argument, on obtient que pour toute géodésique $γ_{1}$ de type lumière coupant γ, $ψ = ψ_{\max}$ en tout point de $γ_{1}$ , et ainsi de suite. On en déduit ψ est constante sur la variété entière, car deux points quelconques peuvent être joints par une suite de géodésiques de type lumière.

Reçu le : 2009-05-14
Accepté le : 2009-06-25
Publié le : 2009-08-04

DOI : 10.1016/j.crma.2009.06.017

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     author = {Volodymyr Kiosak and Vladimir S. Matveev},
     title = {There are no conformal {Einstein} rescalings of complete {pseudo-Riemannian} {Einstein} metrics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1067--1069},
     publisher = {Elsevier},
     volume = {347},
     number = {17-18},
     year = {2009},
     doi = {10.1016/j.crma.2009.06.017},
     language = {en},
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Volodymyr Kiosak; Vladimir S. Matveev. There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1067-1069. doi : 10.1016/j.crma.2009.06.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.017/

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