  A general concept of potential field is introduced. The potential field that
one puts in correspondence with dark matter, has fundamental geometrical
interpretation (parallel transport) and has intrinsically inherent in local
symmetry. The equations of dark matter field are derived that are invariant
with respect to the local transformations. It is shown how to reduce these
equations to the Maxwell equations. Thus, the dark matter field may be
considered as generalized electromagnetic field and a simple solution is given
of the old problem to connect electromagnetic field with geometrical properties
of the physical manifold itself. It is shown that gauge fixing renders
generalized electromagnetic field effectively massive while the Maxwell
electromagnetic field remains massless. To learn more about interactions
between matter and dark matter on the microscopical level (and to recognize the
fundamental role of internal symmetry) the general covariant Dirac equation is
derived in the Minkowski space--time which describe the interactions of spinor
field with dark matter field.
