  The $h$-deformed quantum plane is a counterpart of the $q$-deformed one in
the set of quantum planes which are covariant under those quantum deformations
of GL(2) which admit a central determinant. We have investigated the
noncommutative geometry of the $h$-deformed quantum plane. There is a
2-parameter family of torsion-free linear connections, a 1-parameter sub-family
of which are compatible with a skew-symmetric non-degenerate bilinear map. The
skew-symmetric map resembles a symplectic 2-form and induces a metric. It is
also shown that the extended $h$-deformed quantum plane is a noncommutative
version of the Poincar\'e half-plane, a surface of constant negative Gaussian
curvature.
