  The possible extensions of GR for description of fermions on a curved space,
for supergravity and for loop quantum gravity require a richer set of 16
independent variables. These variables can be assembled in a coframe field,
i.e., a local set of four linearly independent 1-forms. In the ordinary
formulation, the coframe gravity does not have any connection to a specific
geometry even being constructed from the geometrical meaningful objects. A
geometrization of the coframe gravity is an aim of this paper. We construct a
complete class of the coframe connections which are linear in the first order
derivatives of the coframe field on an $n$ dimensional manifolds with and
without a metric. The subclasses of the torsion-free, metric-compatible and
flat connections are derived. We also study the behavior of the geometrical
structures under local transformations of the coframe. The remarkable fact is
an existence of a subclass of connections which are invariant when the
infinitesimal transformations satisfy the Maxwell-like system of equations. In
the framework of the coframe geometry construction, we propose a geometrical
action for the coframe gravity. It is similar to the Einstein-Hilbert action of
GR, but the scalar curvature is constructed from the general coframe
connection. We show that this geometric Lagrangian is equivalent to the coframe
Lagrangian up to a total derivative term. Moreover there is a family of coframe
connections which Lagrangian does not include the higher order terms at all. In
this case, the equivalence is complete.
