Weyl forms in parabolic contact geometries

Abstract

In a parabolic contact geometry a choice of contact form determines a Weyl form consisting of three components: a soldering form, a Weyl connection, and a Rho tensor. An explicit description of the soldering form is well-known. We express the Rho tensor in terms of the Weyl connection in the setting of torsion-free normal parabolic contact geometries. Parabolic contact geometries with a specified contact form have a second canonical connection, the Tanaka-Webster connection. We express the Weyl connection in terms of the Tanaka-Webster connection in the setting of normal parabolic contact geometries. As an application of some of these results we derive formulas for the harmonic curvature components in integrable Legendrian contact geometry, recovering a result of S. Ivanov, D. Vassilev, and S. Zamkovoy.