Asian Journal of Mathematics

Volume 18 (2014)

Number 1

Warped product Einstein metrics over spaces with constant scalar curvature

Pages: 159 – 190



Chenxu He (Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

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

William Wylie (Department of Mathematics, Syracuse University, Syracuse, New York, U.S.A.)


In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.


Einstein manifolds, warped products, rigidity, Ricci solitons, solvable Lie groups

2010 Mathematics Subject Classification

53B20, 53C30

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