Jet Isomorphism for Conformal Geometry

Local invariants of a metric in Riemannian geometry are quantities expressible in local coordinates in terms of the metric and its derivatives and which have an invariance property under changes of coordinates. It is a fundamental fact that such invariants may be written in terms of the curvature tensor of the metric and its covariant derivatives. In this form, they can be identified with invariants of the orthogonal group acting algebraically on the space of possible curvature tensors and derivatives. We refer to the result asserting that the space of infinite order jets of metrics modulo coordinate changes is isomorphic to a space of curvature tensors and derivatives modulo the orthogonal group as a jet isomorphism theorem. Such results recast the study and classification of local geometric invariants in purely algebraic terms, in which form the methods of invariant theory and representation theory can be brought to bear. The goal of this paper is to describe analogous jet isomorphism theorems in the context of conformal geometry. In conformal geometry one is given a metric only up to scale. The results in the conformal case provide a tensorial description of the space of jets of metrics modulo changes of coordinates and conformal factor. The motion group of the flat model is the conformal group G = O(n+1, 1)/{±I} acting projectively on the sphere S and the role of the orthogonal group in Riemannian geometry is played by the parabolic subgroup P ⊂ G preserving a null line. Since P is a matrix group in n+2 dimensions, its natural tensor representations are on tensor powers of R. Thus one expects the appearance of tensors in n+ 2 dimensions in conformal jet isomorphism theorems. When n is odd, the ambient metric construction of [FG1] associates to a conformal Riemannian manifold (M, [g]) of dimension n an infinite order jet of a Lorentzian metric g̃ along a hypersurface in a space G̃ of dimension n+2, uniquely determined up to diffeomorphism. The tensors in the odd-dimensional conformal jet isomorphism theorem are the curvature tensor and its covariant derivatives for the ambient metric. They satisfy extra identities beyond those satisfied by the derivatives of curvature of a general metric as a consequence of the Ricci-flatness and homogeneity conditions satisfied by an ambient metric. The elaboration of these identities and a formulation