Weyl Connections and Curvature Properties of CR Manifolds

Abstract

In this paper we advertise the use of Weyl connections in CRgeometry. To every Weyl connection on a CR manifold weassociate a Tanaka connection and we study its curvature. When theWeyl connection is exact, the corresponding Tanakaconnection is the connection defined by Tanaka (A DifferentialGeometric Study on Strongly Pseudo-Convex Manifolds, 1975).Our more general approach has the advantage that some tediouscalculations which appear in CR geometry, using the ordinaryTanaka connections, can be simplified using our generalised Tanakaconnections.

Let (M, I, H) be a CR manifold with a Weyl connection D (that is, a connection on the line bundle TM/H), and let ∇ be the associated Tanaka connection (see Section 1). Let R ^∇ be the curvature of ∇. We show that the I-invariant part of R ^∇| _H×H satisfies theBianchi identity, and we study the case when it is a Kählercurvature tensor. We also determine the explicit expression of theI-anti-invariant part of R ^∇| _H×H , and we useit to characterise Sasakian CR manifolds between all CR manifolds(see Section 2). In Section 3 we studyhow R ^∇ changes under the change of D. Our approach isparticularily appropriate to this computation (see Lemma 13 and Theorem 14) and it avoids the use ofRiemannian metrics associated to contact forms. To every Tanakaconnection ∇ we associate a Kähler curvature tensor(see Theorem 9) whose principal part is independent ofthe choice of D (see Theorem 14). We thus obtain asimple tensorial definition of the Chern–Moser tensor of a CRmanifold. In particular, this definition makes straightforward theidentification, already observed by Webster, of theChern–Moser tensor of a Sasaki CR manifold with the Bochner tensorof the associated Kähler manifold.