  We compute a Bochner type formula for static three-manifolds and deduce some
applications in the case of positive scalar curvature. We also explain in
details the known general construction of the (Riemannian) Einstein
(n+1)-manifold associated to a maximal domain of a static n-manifold where the
static potential is positive. There are examples where this construction
inevitably produces an Einstein metric with conical singularities along a
codimension-two submanifold. By proving versions of classical results for
Einstein four-manifolds for the singular spaces thus obtained, we deduce some
classification results for compact static three-manifolds with positive scalar
curvature.
