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.

30 pages Latex (one figure available as a postscript file)