Conformal structures associated to generic rank 2 distributions on 5-manifolds – characterization and Killing-field decomposition. (English) Zbl 1191.53016

Hammerl, Matthias; Sagerschnig, Katja

SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 081, 29 p. (2009).

Summary: Given a maximally non-integrable 2-distribution \({\mathcal D}\) on a 5-manifold \(M\), it was discovered by P. Nurowski that one can naturally associate to \({\mathcal D}\) a conformal structure \([g]_{\mathcal D}\) of signature \((2,3)\) on \(M\). We show that these conformal structures \([g]_{\mathcal D}\) obtained by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of \([g]_{\mathcal D}\) can be decomposed into a symmetry of \({\mathcal D}\) and an almost Einstein scale of \([g]_{\mathcal D}\).

MSC:

53A55	Differential invariants (local theory), geometric objects
34A26	Geometric methods in ordinary differential equations
35N10	Overdetermined systems of PDEs with variable coefficients
53A30	Conformal differential geometry (MSC2010)
53B15	Other connections
53B30	Local differential geometry of Lorentz metrics, indefinite metrics
58A30	Vector distributions (subbundles of the tangent bundles)

Keywords:

generic distributions; conformal geometry; tractor calculus; Fefferman construction; conformal Killing fields; almost Einstein scales