1 0 Ju n 20 03 TRANSVERSE RIEMANN - LORENTZ METRICS WITH TANGENT RADICAL

Consider a smooth manifold with a smooth metric which changes bilinear type from Riemann to Lorentz on a hypersurface Σ with radical tangent to Σ. Two natural bilinear symmetric forms appear there, and we use it to analyze the geometry of Σ. We show the way in which these forms control the smooth extensibility over Σ of the covariant, sectional and Ricci curvatures of the Levi-Civita connection outside Σ. 1 Preliminaries Let M be a m-dimensional connected manifold (m > 2) endowed with a smooth, symmetric (0, 2)-tensorfield g which fails to have maximal rank on a (non void) subset Σ ⊂ M. Thus, at each point p ∈ Σ, there exists a nontrivial subspace (the radical) Rad p ⊂ T p M, which is orthogonal to the whole T p M. We say that (M, g) is a singular space (these spaces were analyzed for the first time in [7]). Moreover we say that (M, g) is a transverse singular space if, for any local coordinate system (x has non-zero differential at the points of Σ (i.e. where det(g ab) vanishes). This implies at once: (1) the subset Σ is a smooth hypersurface in M, called the singular hypersurface; (2) at each point p ∈ Σ the radical Rad p is one