Natural operations on differential forms on contact manifolds.pdfVIP

Di?erential Geometry and its Applications 50 (2017) 34–51 Contents lists available at ScienceDirect Di?erential Geometry and its Applications /locate/difgeo Natural operations on di?erential forms on contact manifolds Andreas Bernig 1 Institut für Mathematik, Goethe-Universit?t Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany article info Article history: Received 6 September 2016 Available online 16 November 2016 Communicated by J. Slovák MSC: 53D10 58A10 Keywords: Natural operator Contact manifold Rumin di?erential abstract We characterize all natural linear operations between spaces of di?erential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior derivative. ? 2016 Elsevier B.V. All rights reserved. 1. Introduction A classical theorem due to Palais [20] characterizes those linear operations on di?erential forms on a manifold which are compatible with di?eomorphisms. The result is roughly that only the identity and the exterior di?erential have these properties. The linearity assumption was removed by Kolá?–Michor–Slovák in 1993 [18]. A very recent result by Navarro–Sancho [19] generalizes this theorem further by considering natural operations on k-tuples of di?erential forms. They prove that such operations can be written as polynomials in the given forms and their exterior di?erentials. A special case of this theorem was shown earlier by Freed–Hopkins [13]. A natural general question in this context is the following. Assume that M is endowed with some extra structure (for instance, a contact or symplectic structure, an almost complex structure etc.). What are the operations on di?erential forms which are compatible with the di?eomorphisms of M respecting the extra structure? In the present paper, we study this question for contact manifolds and contactomorphisms. Before describing more precisely our result, let us recall the de?nition of contact mani