pseudo-Riemannian Einstein Metric Research Articles

New pseudo Einstein metrics on Einstein solvmanifolds

A Riemannian Einstein manifold is called an Einstein solvmanifold if there exists a transitive solvable group of isometries. In this short note, we show that every Einstein solvmanifold admits at least one pseudo-Riemannian Einstein metric.

Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces

The aim of this work is to study homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. First, we deduce a formula for Ricci tensor of a homogeneous pseudo-Riemannian manifold with compact isotropy subgroup. Based on this formula, we establish a one-to-one correspondence between homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces and homogeneous Riemannian Einstein metrics on compact homogeneous spaces. As an application, we prove that every noncompact connected simple Lie group except SL(2) admits at least two nonproportional left invariant pseudo-Riemannian Einstein metrics. Furthermore, we study left invariant pseudo-Riemannian Einstein metrics on solvable Lie groups. By showing that if a nilpotent Lie group admits a left invariant Riemannian Ricci soliton, then it admits a left invariant pseudo-Riemannian Ricci soliton as well, we construct a left invariant pseudo-Riemannian Einstein metric on any given Riemannian standard Einstein solvmanifold.

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

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).

Ricci-flat and Einstein pseudoriemannian nilmanifolds

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

Indefinite Einstein metrics on simple Lie groups

The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein metrics on a given semisimple Lie group always includes D, the Levi-Civita connection of the Killing form. For the groups SU(l,j) (or SL(n,R), or SL(n,C) or, if n is even, SL(n/2,IH)), with 0<=j<=l and j+l>2 (or, n>2), we explicitly describe the connected component C of E, containing D. It turns out that C, a relatively-open subset of E, is also an algebraic variety of real dimension 2lj (or, real/complex dimension [n^2/2] or, respectively, real dimension 4[n^2/8]), forming a union of (j + 1)(j + 2)/2 (or, [n^2]+1 or, respectively, [n/4] + 1) orbits of the adjoint action. In the case of SU(n) one has 2lj=0, so that a positive-definite multiple of the Killing form is isolated among suitably normalized left-invariant Riemannian Einstein metrics on SU(n).

Einstein metrics in projective geometry

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.

Einstein metrics on group manifolds and cosets

It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under G L × G R . Less well known is that every compact simple group manifold except S O ( 3 ) and S U ( 2 ) admits at least one more homogeneous Einstein metric, invariant still under G L but with some, or all, of the right-acting symmetry broken. ( S O ( 3 ) and S U ( 2 ) are exceptional in admitting only the one, bi-invariant, Einstein metric.) In this paper, we look for Einstein metrics on three relatively low-dimensional examples, namely G = S U ( 3 ) , S O ( 5 ) and G 2 . For G = S U ( 3 ) , we find just the two already known inequivalent Einstein metrics. For G = S O ( 5 ) , we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G = G 2 we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G / H for the above groups G . In particular, for S O ( 5 ) / U ( 1 ) we find, depending on the embedding of the U ( 1 ) , generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature ( 2 , 6 ) on S U ( 3 ) , an Einstein metric of signature ( 5 , 6 ) on G 2 / S U ( 2 ) diag , and an Einstein metric of signature ( 4 , 6 ) on G 2 / U ( 2 ) . Interestingly, there are no Lorentzian Einstein metrics among our examples.

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

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 ⋅ 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. To cite this article: V. Kiosak, V.S. Matveev, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Generalized Goldberg-Sachs theorems for pseudo-Riemannian four-manifolds

It has been recently observed that the generalized Goldberg-Sachs theorem in general relativity as well as some of its corollaries admit appropriate Riemannian versions. In this paper we use the formalism of spinors to give alternative proofs of these results clarifying the analogy between positive Hermitian structures of oriented Riemannian four-manifolds and shear-free congruences of oriented Lorentzian four-manifolds. We also prove similar results for oriented pseudo-Riemannian four-manifolds when the metric is of zero signature. This allows us to describe compact oriented four-manifolds possibly admitting a pseudo-Riemannian Einstein metric of zero signature whose positive Weyl tensor has two distinct eigenvalues corresponding to non-isotropic eigenspaces.