1 / 12

On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

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 : degenerates }≠ ø Rad p (M):={ : } ≠ø, p ∑

hazina
Télécharger la présentation

On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  3. 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

  4. 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

  5. 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?

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

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

  8. 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.

  9. 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

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

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

  12. 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

More Related