Lightlike manifolds and Cartan geometries
| Autor: | Francisco J. Palomo |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
Physics Pure mathematics Algebra and Number Theory Spacetime 010102 general mathematics 01 natural sciences Pseudo-Riemannian manifold Manifold General Relativity and Quantum Cosmology symbols.namesake Hypersurface Differential Geometry (math.DG) Cartan connection 0103 physical sciences FOS: Mathematics symbols Connection form Mathematics::Differential Geometry 010307 mathematical physics 0101 mathematics Conformal geometry Mathematical Physics Analysis Distribution (differential geometry) |
| Zdroj: | Analysis and Mathematical Physics. 11 |
| ISSN: | 1664-235X 1664-2368 |
| DOI: | 10.1007/s13324-021-00547-8 |
| Popis: | Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold $N$ provides a lightlike metric $h$ with radical distribution globally spanned by a vector field $Z$. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold $(N,h)$, we show a method to construct a family of ambient Lorentzian manifolds that realize $(N,h)$ as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when $(N,h)$ is generic. |
| Databáze: | OpenAIRE |
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