A note on static spaces and related problems

In this paper we study static spaces introduced in Hawking and Ellis (1975)  1], Fischer and Marsden (1975) 3] and Riemannian manifolds possessing solutions to the critical point equation introduced in Besse (1987)  11], Hwang (2000)  12]. 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)  3] of classifying vacuum static spaces and in proving the conjecture proposed in Besse (1987)  11] concerning manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.