Ultrastatic
Ultrastatic spacetimes are static spacetimes (with a timelike Killing vector that is hypersurface orthogonal) for which the Killing vector is also of constant norm on the hypersurface, so that it does not depend on the point in space.
1. History
2. Topology
3. Metrics and coordinates
The metric of an ultrastatic spacetime can simply be constructed as the product of a Riemannian metric with $\mathbb{R}$
$$ds^2 = -dt^2 + h$$