2016/07/12 by Wolfgang Kühnel, Kühnel, Wolfgang, Hans–Bert Rademacher +1 · 2 citations
Mathematics · Medicine · #53C25 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Pelvic and Acetabular Injuries
paper · pdf · doi:10.48550/arxiv.1607.03332
openalex publication_date 2016/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 4-dimensional 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.