A note on static spaces and related problems
Published Web Locationhttps://doi.org/10.1016/j.geomphys.2013.07.003
In this paper we study static spaces introduced in Hawking and Ellis (1975) , Fischer and Marsden (1975)  and Riemannian manifolds possessing solutions to the critical point equation introduced in Besse (1987) , Hwang (2000) . In both cases, on the manifolds there is a function satisfying a particular Ricci-Hessian type equation (1.6). With an idea similar to that used in Cao et al. (2012) [15,16], we have made progress in solving the problem raised in Fischer and Marsden (1975)  of classifying vacuum static spaces and in proving the conjecture proposed in Besse (1987)  concerning manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3. © 2013 Elsevier B.V.