Conformally Einstein product spaces

Abstract

We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these manifolds is 1-dimensional and the case where the conformal factor depends on both manifolds simultaneously. If both factors are at least 3-dimensional then the latter case reduces to the product of two Einstein spaces, each of the special type admitting a non-trivial conformal gradient field. These are completely classified. If each factor is 2-dimensional, there is a special family of examples of non-constant curvature (called extremal metrics by Calabi), where in each factor the gradient of the Gaussian curvature is a conformal vector field. Then the metric of the 2-manifold is a warped product where the warping function is the first derivative of the Gaussian curvature. Moreover we find explicit examples of Einstein warped products with a 1-dimensional fibre and such with a 2-dimensional base. Therefore in the 4dimensional case our Main Theorem points towards a local classification of conformally Einstein products. Finally we prove an assertion in the book by A.Besse on complete Einstein warped products with a 2-dimensional base. All solutions can be explicitly written in terms of integrals of elementary functions.

Conformally related Riemannian metrics with non-generic holonomy

“…thus proving that ϕ is constant on M if n ≥ 3. The same conclusion holds for n = 2, since in this case (8) shows directly that ϕ is harmonic, thus constant.…”

“…This formula shows that the constant Scalg is non-negative at a point where ϕ attains its maximum, and non-positive at a point where ϕ attains its minimum. Thus Scalg = 0, so (8) ∆ g ϕ = (n − 2)|dϕ| 2 g . Integrating this equation over M with respect to the volume form dµ g we obtain…”

“…In this case (M, g) has, up to a finite covering, two parallel orthogonal distributions, whose volume forms define conformal Killing formsω 1 andω 2 on the irreducible manifold (M, g). Moreover, a result of Cleyton [5], also proved by Kühnel and Rademacher [8], shows that the conformal change of the metric only depends on one of the factors. This implies that eitherω 1 orω 2 is actually a Killing form on (M,g) and one can apply the classification of Killing forms on manifolds with special holonomy obtained in [2] for locally symmetric spaces, in [12] for quaternionic-Kähler manifolds, and in [18] for G 2 and Spin (7) manifolds.…”

Conformal geometry of non‐reductive four‐dimensional homogeneous spaces

“…It follows from (17) and (18) Since a = 0 and b = −q the first two equations in (50) show that ϕ does not depend on the coordinates x 1 and x 2 and the tensor field C reduces to b ϕ C 344 = −(b + q) (ϕ − ϕ 3 ) , where ϕ is a smooth function on the coordinates (x 3 , x 4 ). Now C 344 = 0 gives ϕ(x 3 , x 4 ) = φ(x 4 )e x 3 , for some smooth function φ(x 4 ).…”

“…In dimension four the Bach flat equation is only a necessary condition since there are Bach flat spaces which are not conformally Einstein. An important class of four-dimensional conformally Einstein metrics is obtained by considering the product of surfaces with nowhere zero scalar curvature under some additional conditions (see the survey [16,17] for more information). Moreover in such cases the conformal Einstein metric is unique up to a constant.…”