  The scalar invariant, I, constructed from the "square" of the first covariant
derivative of the curvature tensor is used to probe the local geometry of
static spacetimes which are also Einstein spaces. We obtain an explicit form of
this invariant, exploiting the local warp-product structure of a 4-dimensional
static spacetime, $~^{(3)}\Sigma \times_{f} \reals$, where $^{(3)}\Sigma $ is
the Riemannian hypersurface orthogonal to a timelike Killing vector field with
norm given by a positive function, $f$ on $^{(3)}\Sigma $. For a static
spacetime which is an Einstein space, it is shown that the locally measurable
scalar, I, contains a term which vanishes if and only if $^{(3)}\Sigma$ is
conformally flat; also, the vanishing of this term implies (a) $~^{(3)}\Sigma$
is locally foliated by level surfaces of $f$, $^{(2)}S$, which are totally
umbilic spaces of constant curvature, and (b) $^{(3)}\Sigma$ is locally a
warp-product space. Futhermore, if $^{(3)}\Sigma$ is conformally flat it
follows that every non-trivial static solution of the vacuum Einstein equation
with a cosmological constant, is either Nariai-type or Kottler-type - the
classes of spacetimes relevant to quantum aspects of gravity.
