Mathematics
Conformally Kähler base metrics for Einstein warped products
Document Type
Article
Abstract
A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (a/f)df+r=g, for a smooth nonconstant function f and constants and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP1 bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form. © 2010 Elsevier B.V.
Publication Title
Differential Geometry and its Application
Publication Date
2-2011
Volume
29
Issue
1
First Page
85
Last Page
92
ISSN
0926-2245
DOI
10.1016/j.difgeo.2010.12.001
Keywords
conformal, Kähler, primary, Quasi-Einstein, secondary, warped product
Repository Citation
Maschler, Gideon, "Conformally Kähler base metrics for Einstein warped products" (2011). Mathematics. 37.
https://commons.clarku.edu/mathematics/37
