Conformally warped manifolds and quasi-Einstein metrics

Details Conformally warped manifolds and quasi-Einstein metrics Conformally warped manifolds and quasi-Einstein metrics

2010-11-11

arxiv.org/abs/1011.2723v2

Mathematics

Differential Geometry

41 pages; clarified a statement and added a reference

Scientific paper

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-\'Emery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons, and static metrics, all of which are very important within their fields. In this article, we introduce the notion of conformally warped manifolds (CWMs) and quasi-Einstein metrics on them as a means to unite these perspectives. We will offer many results and interpretations which illustrate the unifying nature of CWMs, including a natural variational description of quasi-Einstein metrics as well as some interesting families of such metrics.

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