Communications in Analysis and Geometry

Volume 20 (2012)

Number 2

On the classification of warped product Einstein metrics

Pages: 271 – 311



Chenxu He (Department of Mathematics, University of Pennsylvania)

Peter Petersen (Department of Mathematics, University of California at Los Angeles)

William Wylie (Department of Mathematics, Syracuse University, Syracuse, New York)


In this paper we take the perspective introduced by Case–Shu–Wei ofstudying warped product Einstein metrics through the equation forthe Ricci curvature of the base space. They call this equation onthe base the $m$-quasi Einstein equation, but we will also call itthe $(\lambda,n+m)$-Einstein equation. In this paper we extend thework of Case–Shu–Wei and some earlier work of Kim–Kim to allow thebase to have non-empty boundary. This is a natural case to considersince a manifold without boundary often occurs as a warped productover a manifold with boundary, and in this case we get some interestingnew canonical examples. We also derive some new formulas involvingcurvatures that are analogous to those for the gradient Ricci solitons.As an application, we characterize warped product Einstein metricswhen the base is locally conformally flat.

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