Conformally Kähler base metrics for Einstein warped products

الأصل · EN

A Riemannian metric g with Ricci curvature is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies (-a/f) df+=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, by a result of Kim and Kim, such a metric forms a base for certain warped Einstein metrics. On a manifold M of real dimension at least six, let (g,) be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant Killing potential. Suppose the metric g=g/² is nontrivial on M⁻¹(0), and the associated function f is locally a function of. Then (g,) is an pair, a notion defined by Derdzinski and Maschler. This implies that M is biholomorphic to an open set in the total space of a CP¹ bundle whose base manifold admits a Kähler-Einstein metric. If M is additionally compact, it is a total space of such a bundle or complex projective space. Also, the function f is affine in ⁻¹ with nonzero constants. Conversely, in all even dimensions n≥ 4, there exist pairs (g,) and corresponding nonzero constants K and L for which g/² is nontrivial quasi-Einstein with f=K⁻¹+L. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler is reproduced in dimension at least six in a more explicit form.