  We consider the spacetime geometry of a static but otherwise generic black
hole (that is, the horizon geometry and topology are not necessarily
spherically symmetric). It is demonstrated, by purely geometrical techniques,
that the curvature tensors, and the Einstein tensor in particular, exhibit a
very high degree of symmetry as the horizon is approached. Consequently, the
stress-energy tensor will be highly constrained near any static Killing
horizon. More specifically, it is shown that -- at the horizon -- the
stress-energy tensor block-diagonalizes into ``transverse'' and ``parallel''
blocks, the transverse components of this tensor are proportional to the
transverse metric, and these properties remain invariant under static conformal
deformations. Moreover, we speculate that this geometric symmetry underlies
Carlip's notion of an asymptotic near-horizon conformal symmetry controlling
the entropy of a black hole.
