  We classify the metric-affine theories of gravitation, in which the metric
and the connections are treated as independent variables, by use of several
constraints on the connections. Assuming the Einstein-Hilbert action, we find
that the equations for the distortion tensor (torsion and non-metricity) become
algebraic, which means that those variables are not dynamical. As a result, we
can rewrite the basic equations in the form of Riemannian geometry. Although
all classified models recover the Einstein gravity in the Palatini formalism
(in which we assume there is no coupling between matter and the connections),
but when matter field couples to the connections, the effective Einstein
equations include an additional hyper energy-momentum tensor obtained from the
distortion tensor. Assuming a simple extension of a minimally coupled scalar
field in metric-affine gravity, we analyze an inflationary scenario. Even if we
adopt a chaotic inflation potential, certain parameters could satisfy
observational constraints. Furthermore, we find that a simple form of Galileon
scalar field in metric-affine could cause G-inflation.
