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 $D^{\mathrm{\infty }}$. Suppose that the radical annihilator hyperplane is tangent to $D^{\mathrm{\infty }}$. We examine the geometry of the ($g^{\ast }$-dual) covariant metric $g$ on $M-$ $D^{\mathrm{\infty }}$, prove the existence of a canonical (polar-normal) vectorfield whose integral curves are $C^{\mathrm{\infty }}-$pregeodesics crossing $D^{\mathrm{\infty }}$ 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.