On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds V. Fernández Universidad Complutense de Madrid

Transverse Riemann-Lorentz Manifolds M connected manifold symmetric (0,2) tensor field on M symmetric (0,2) tensor field on M ∑:={ pM : degenerates}≠ø Rad p (M):={ : }≠ø, p∑ M-∑ is a union of pseudoriemannian manifolds

DEF: M is a transverse type-changing manifold if for every p∑ ∑ is a hypersurface that locally separates two open pseudoriemannian manifolds whose indices differ in one and dim(Rad p (M))=1

DEF: M is a transverse Riemann-Lorentz manifold if the components of M-∑ are either riemannian or lorentzian. Example: ∑ : type-changing hypersurface M + : riemannian M - : lorentzian

On M-∑ we have all the geometric objects associated to g derived from the Levi-Civita connection D: covariant derivatives, parallel transport, geodesics, curvatures… PROBLEM: Extendability to M of these objects?

□ the unique torsion-free and metric dual connection on Msuch that on M-∑ □ the unique torsion-free and metric dual connection on M such that on M-∑ R a radical vectorfield (R p Rad p (M)-{0}) well-defined map

DEF: M is II-FLAT if OBS: M is II-flat iff extends to M whenever or are tangent to ∑. DEF: M is III-FLAT if

THEOREM: (Kossowski,97) K covariant curvature extends to M iff radical transverse to ∑ and M II-flat. Ric Ricci tensor extends to M iff radical transverse to ∑ and M III-flat.

Conformal Geometry transverse Riemann-Lorentz transverse Riemann-Lorentz C= conformal structure is also transverse Riemann-Lorentz whith the same type-changing hypersurface ∑ and same radical Rad DEF: conformal transverse Riemann- Lorentz manifold conformal transverse Riemann- Lorentz manifold

Weyl Conformal Curvature DEF: N pseudoriemannian manifold Weyl tensor of N, where Schouten tensor Schouten tensor Kulkarni-Nomizu product Kulkarni-Nomizu product

OBS: thus conformal invariant, called Weyl conformal curvature THEOREM: (Weyl, 1918) iff M conformally flat iff M conformally flat that is, around every pN there exists a metric on the conformal structure which is flat

Extendability of Weyl tensor THEOREM: W extends to the whole M iff radical transverse and M conformally III-flat that is, around every p∑ there exists a metric on the conformal structure which is III-flat