  The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the
Finsler-like geometry (in the sense of d-connection, d-torsions and
d-curvatures) of the rheonomic Berwald-Moor metric. A natural geometrical
gravitational field theory produced by the rheonomic Berwald-Moor metric is
also constructed.
