Mathematics – Differential Geometry
Scientific paper
2008-07-05
Mathematics
Differential Geometry
This version replaces the first one, and its main purpose is to correct Theorem A. This version has 11 pages
Scientific paper
A Riemannian metric $\wht{g}$ with Ricci curvature $\wht{\ri}$ is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies $(-a/f)\wht{\nab} df+\wht{\ri}=\lambda \wht{g}$, for a smooth nonconstant function $f$ and constants $\lambda$ 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 $(g,\t)$ be a pair consisting of a K\"ahler metric $g$ which is locally K\"ahler irreducible, and a nonconstant Killing potential $\t$. Suppose the metric $\wht{g}=g/\t^2$ is nontrivial \bee on $M\setminus\t^{-1}(0)$, and the associated function $f$ is locally a function of $\t$. Then $(g,\t)$ is an \sk\ pair, a notion defined by Derdzinski and Maschler. This implies that $M$ is biholomorphic to an open set in the total space of a $CP^1$ bundle whose base manifold admits a K\"ahler-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 $\t^{-1}$ with nonzero constants. Conversely, in all even dimensions $n\geq 4$, there exist \sk pairs $(g,\t)$ and corresponding nonzero constants $K$ and $L$ for which $g/\t^2$ is nontrivial quasi-Einstein with $f=K\t^{-1}+L$. Additionally, a result of Case, Shu and Wei on the K\"ahler reducibility of nontrivial K\"ahler \bers is reproduced in dimension at least six in a more explicit form.
No associations
LandOfFree
Conformally Kähler base metrics for Einstein warped products does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Conformally Kähler base metrics for Einstein warped products, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Conformally Kähler base metrics for Einstein warped products will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-448809