Static
Static spacetimes are spacetimes containing one timelike Killing vector, and can be recast in a way independant of a time coordinate.
1. History
2. Topology
The topologies allowed for a static spacetime are fairly broad but it requires a foliation by spacelike hypersurfaces, forbidding for instance topology changes.
3. Metrics and coordinates
Static spacetimes admit metrics of the form
$$ds^2 = - e^{2f(x)} dt^2 + h(x)$$with $f$ a function depending only on the spacelike coordinates and $h$ a Riemannian metric on the spacelike hypersurface that does not depend on the coordinate $t$. The tangent vector $\partial_t$ is a timelike Killing vector.