  In this work we provide a possible geometrical interpretation of the spin of
elementary particles. In particular, it is investigated how the wave equations
of matter are altered by the addition of an antisymmetric contribution to the
metric tensor. In this scenario the explicit form of the matter wave equations
is investigated in a general curved space-time, and then the equations are
particularized to the flat case. Unlike traditional approaches of NGT, in which
the gravitational field is responsible for breaking the symmetry of the flat
Minkowski metric, we find more natural to consider that, in general, the metric
of the space-time could be nonsymmetric even in the flat case. The physical
consequences of this assumption are explored in detail. Interestingly enough,
it is found that the metric tensor splits into a bosonic and a fermionic; the
antisymmetric part of the metric is very sensitive to the spin and turns out to
be undetectable for spinless scalar particles. However, fermions couple to it
in a non-trivial way (only when there are interactions). In addition, the Pauli
coupling is derived automatically as a consequence of the nonsymmetric nature
of the metric
