Conformal vector fields and non-degenerate distributions

Abstract

Given a manifold with a conformal metric [g] and a non-degenerate proper distribution W we construct a unique connection, with natural restriction on the torsion, which preserves both [g] and W. Moreover, this connection (which is essentially a Weyl connection with torsion) is preserved by any conformal isometry preserving W. In particular the result has relevance for the linearizability problem of conformal vector fields, for any such vector field preserving W becomes affine and is thus linearized about any fixed point by exponential coordinates. Further applications involving spacetimes are highlighted.