  Equations of divergence type in static spacetimes play a significant role in
the proof of uniqueness theorems of black holes. We generalize the divergence
equation originally discovered by Robinson in four dimensional vacuum
spacetimes into several directions. We find that the deviation from spherical
symmetry is encoded in a symmetric trace-free tensor $H_{ij}$ on a static
timeslice. This tensor is the crux for the construction of the desired
divergence equation, which allows us to conclude the uniqueness of the
Schwarzschild black hole without using Smarr's integration mass formula. In
Einstein-Maxwell(-dilaton) theory, we apply the maximal principle for a number
of divergence equations to prove the uniqueness theorem of static black holes.
In higher $(n\ge 5)$ dimensional vacuum spacetimes, a central obstruction for
applicability of the current proof is the integration of the
$(n-2)$-dimensional scalar curvature over the horizon cross-section, which has
been evaluated to be a topological constant by the Gauss-Bonnet theorem for
$n=4$. Nevertheless, it turns out that the $(n-1)$-dimensional symmetric and
traceless tensor $H_{ij}$ is still instrumental for the modification of the
uniqueness proof based upon the positive mass theorem, as well as for the
derivation of the Penrose-type inequality.
