  Ashtekar's formulation for canonical quantum gravity is known to possess the
topological solutions which have their supports only on the moduli space $\CN$
of flat $SL(2,C)$ connections. We show that each point on the moduli space
$\CN$ corresponds to a geometric structure, or more precisely the Lorentz group
part of a family of Lorentzian structures, on the flat (3+1)-dimensional
spacetime. A detailed analysis is given in the case where the spacetime is
homeomorphic to $R\times T^{3}$. Most of the points on the moduli space $\CN$
yield pathological spacetimes which suffers from singularities on each spatial
hypersurface or which violates the strong causality condition. There is,
however, a subspace of $\CN$ on which each point corresponds to a family of
regular spacetimes.
