Skew-symmetric endomorphisms in M1,3 : a unified canonical form with applications to conformal geometry

Abstract

Abstract We derive a canonical form for skew-symmetric endomorphisms F in Lorentzian vector spaces of dimension three and four which covers all non-trivial cases at once. We analyze its invariance group, as well as the connection of this canonical form with duality rotations of two-forms. After reviewing the relation between these endomorphisms and the algebra of conformal Killing vectors of S 2 , C K i l l S 2 , we are able to also give a canonical form for an arbitrary element ξ ∈ C K i l l S 2 along with its invariance group. The construction allows us to obtain explicitly the change of basis that transforms any given F into its canonical form. For any non-trivial ξ we construct, via its canonical form, adapted coordinates that allow us to study its properties in depth. Two applications are worked out: we determine explicitly for which metrics, among a natural class of spaces of constant curvature, a given ξ is a Killing vector and solve all local traceless and transverse tensors that satisfy the Killing initial data equation for ξ. In addition to their own interest, the present results will be a basic ingredient for a subsequent generalization to arbitrary dimensions.