Transverse Riemann-Lorentz type-changing metrics with polar end

Consider a smooth manifold $M$ with a smooth cometric $g^{\ast}$ which changes the bilineal type by transverse way, on a hypersurface $D^{\infty}$. Suppose that the radical annihilator hyperplane is tangent to $D^{\infty}$. We examine the geometry of the ($g^{\ast}$-dual) covariant metric $g$ on $M-$ $D^{\infty}$, prove the existence of a canonical (polar-normal) vectorfield whose integral curves are $C^{\infty}-$pregeodesics crossing $D^{\infty}$ transversely for each point, and analyze the curvature behavior using a natural coordinates. Finally we give an approach to the conformal geometry of such spaces and suggest some application as cosmological big-bang model.