  In this paper we extend the local scalar curvature rigidity result in [6] to
a small domain on general vacuum static spaces, which confirms the interesting
dichotomy of local surjectivity and local rigidity about the scalar curvature
in general in the light of the paper [10]. We obtain the local scalar curvature
rigidity of bounded domains in hyperbolic spaces. We also obtain the global
scalar curvature rigidity for conformal deformations of metrics in the domains,
where the lapse functions are positive, on vacuum static spaces with positive
scalar curvature, and show such domains are maximal, which generalizes the work
in [15].
