  Although black holes are eminent manifestations of very strong gravity, the
geometry of space-time around and even inside them can be significantly
affected by additional bodies present in their surroundings. We study such an
influence within static and axially symmetric (electro-)vacuum space-times
described by exact solutions of Einstein's equations, considering
astrophysically motivated configurations (such as black holes surrounded by
rings) as well as those of pure academic interest (such as specifically "tuned"
systems of multiple black holes). The geometry is represented by the simplest
invariants determined by the metric (the lapse function) and its gradient
(gravitational acceleration), with special emphasis given to curvature (the
Kretschmann and Ricci-square scalars). These quantities are analyzed and their
level surfaces plotted both above and below the black-hole horizons, in
particular near the central singularities. Estimating that the black hole could
be most strongly affected by the other black hole, we focus, in this first
paper, on the Majumdar--Papapetrou solution for a binary black hole and compare
the deformation caused by "the other" hole (and the electrostatic field) with
that induced by rotational dragging in the well-known Kerr and Kerr--Newman
solutions.
