A Calculus for Differential Invariants of Parabolic Geometries

The Wünsch’s calculus for conformal Riemannian invariants is reformulated and essentially generalized to all parabolic geometries. Our approach is based on the canonical Cartan connections and the Weyl connections underlying all such geometries. The differential invariants for various geometric structures are the core ingredients for numerous applications both in geometry and geometric analysis. The invariants of conformal Riemannian manifolds themselves attracted a lot of attention in the course of the last 100 years. In the differential geometry of manifolds equipped with a linear connection, the so called ‘first invariant theorem’ says that all the invariants are expressions built of the curvature and the covariant derivatives of the connection by means of algebraic tensorial invariants. Let us call them the affine differential invariants. The analogous ‘first invariant theorem’ for Riemannian geometries says that all differential invariants are built of the affine invariants of the canonical Levi–Civita connection with additional help of the orthogonal algebraic invariants. Thus, the construction of all invariants is described easily in principle, but the difficult questions on the relations between the individual invariants remains. A conformal Riemannian geometry is defined as the class of conformal Riemannian metrics and so the above Riemannian invariant theorem can be reflected in the definition of the conformal invariance. Thus, a conformal invariant is a Riemannian invariant in terms of a metric from the conformal class, such that any change of the metric leaves its values unchanged. Another equivalent definition of the conformal structures treats them in terms of classical G–structures as reductions of the linear frame bundle to the structure group G0, the group of all conformal Riemannian transformations in the given dimension. This is also equivalent to the choice of a class of linear connections without torsion whose structure group is the conformal Riemannian group. Such a class of connections is always parameterized by one– forms and the Levi–Civita connections coming from the metrics in the conformal class form a subclass parameterized by exact one–forms. The treatment of conformal Riemannian invariants goes back to Cartan, Thomas, Schouten, and others (see e.g. [10, 18, 19]). The broader class of the conformal connections was exploited by Weyl. A lot of spectacular tricks to build invariant expressions have been developed, and some of them were turned into a quite effective calculus for conformal invariants by Wünsch in a series of papers (see e.g. [20]). Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces G/P with G semisimple and P parabolic, the Weyl structures and the preferred connections were introduced in this general framework in [7]. In particular, the notions of scales, closed and exact Weyl connections, and (Schouten’s) Rho–tensors were extended, Research supported by the Institute for Mathematics and its Applications (IMA), University of Minnesota, the first author supported by ‘Fond ???’, the second author supported by the grant