Lie contact structures and conformal structures (Q808442)

The author continues investigations of Lie contact structures [ibid., 13- 41 (1991; see the review above)] and computes the curvature K of the Tanaka connection associated to the Lie contact structure on a unit tangent bundle \(T_ 1M\) of a Riemannian manifold M. This curvature is strictly related to Weyl's conformal curvature of M. When \(K=0\), \(T_ M\) is said to be flat; in this case the structure is locally equivalent to that on the model space \(T_ 1{\mathbb{S}}^ n\). It is proved that if \(\tilde f:\) \(T_ 1M\to T_ 1M'\) is a bundle map preserving the Lie curvature K, then the induced map f: \(M\to M'\) preserves the conformal curvature. Especially, the Lie flatness of the Tanaka connection of \(T_ 1M\) can be characterized by the conformal flatness of M.