On the Conformal Geometry of Transverse RiemannLorentz Manifolds

Transverse Riemann-Lorentz Manifolds M connected manifold symmetric (0, 2) tensor field on M ∑:

DEF: M is a transverse type-changing manifold if for every p ∑ ∑ is

DEF: M is a transverse Riemann-Lorentz manifold if the components of M-∑ are either

On M-∑ we have all the geometric objects associated to g derived from the

□ the unique torsion-free and metric dual connection on M such that on M-∑

DEF: M is II-FLAT if OBS: M is II-flat iff extends to M whenever

THEOREM: (Kossowski, 97) K covariant curvature extends to M iff radical transverse to ∑

Conformal Geometry transverse Riemann-Lorentz C= conformal structure is also transverse Riemann-Lorentz whith the same

Weyl Conformal Curvature DEF: N pseudoriemannian manifold Weyl tensor of N, where Schouten tensor

OBS: thus conformal invariant, called Weyl conformal curvature THEOREM: (Weyl, 1918) iff M conformally

Extendability of Weyl tensor THEOREM: W extends to the whole M iff radical transverse

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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 ∑: ={ p M : Radp(M): ={ degenerates}≠ø : }≠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(Radp(M))=1

DEF: M is a transverse Riemann-Lorentz manifold if the components of M-∑ are either riemannian or lorentzian. Example: ∑: M +: M -: type-changing hypersurface riemannian 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 M such that on M-∑ R a radical vectorfield (Rp Radp(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. Ricci tensor extends to M iff radical transverse to ∑ and M III-flat.

Conformal Geometry 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

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

OBS: thus conformal invariant, called Weyl conformal curvature THEOREM: (Weyl, 1918) 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