  Some basic theorems on Killing vector fields are reviewed. In particular, the
topic of a constant-curvature space is examined. A detailed proof is given for
a theorem describing the most general form of the metric of a homogeneous
isotropic space-time. Although this theorem can be considered to be commonly
known, its complete proof is difficult to find in the literature. An example
metric is presented such that all its spatial cross sections correspond to
constant-curvature spaces, but it is not homogeneous and isotropic as a whole.
An equivalent definition of a homogeneous and isotropic space-time in terms of
embedded manifolds is also given.
