  The fundamental concepts of Riemannian geometry, such as differential forms,
vielbein, metric, connection, torsion and curvature, are generalized in the
context of non-commutative geometry. This allows us to construct the
Einstein-Hilbert-Cartan terms, in addition to the bosonic and fermionic ones in
the Lagrangian of an action functional on non-commutative spaces. As an
example, and also as a prelude to the Standard Model that includes
gravitational interactions, we present a model of chiral spinor fields on a
curved two-sheeted space-time with two distinct abelian gauge fields. In this
model, the full spectrum of the generalized metric consists of pairs of tensor,
vector and scalar fields. They are coupled to the chiral fermions and the gauge
fields leading to possible parity violation effects triggered by gravity.
