  One of the most appealing results of metric-affine gauge theory of gravity is
a close parallel between the Riemann curvature two-form and the Cartan torsion
two-form: While the former is the field strength of the Lorentz-group
connection one-form, the latter can be understood as the field strength of the
coframe one-form. This parallel, unfortunately, is not fully established until
one adopts Trautman's idea of introducing an affine-vector-valued zero-from,
the meaning of which has not been satisfactorily clarified. This paper aims to
derive this parallel from first principles without any ad hoc prescriptions. We
propose a new mathematical framework of an associated affine-vector bundle as a
more suitable arena for the affine group than a conventional vector bundle, and
rigorously derive the covariant derivative of a local section on the
affine-vector bundle in the formal Ehresmann-connection approach. The parallel
between the Riemann curvature and the Cartan torsion arises naturally on the
affine-vector bundle, and their geometric and physical meanings become
transparent. The clear picture also leads to a conjecture about a kinematical
effect of the Cartan torsion that in principle can be measured \`{a} la the
Aharonov-Bohm effect.
