Transverse curves and chain curvature in the Heisenberg group

A chain is the intersection of a complex totally geodesic subspace in complex hyperbolic 2-space with the boundary. The boundary admits a canonical contact structure, and chains are distiguished curves transverse to this structure. The space of chains is analyzed both as a quotient of the contact bundle, and as a subset of ℂP². The space of chains admits a canonical, indefinite Hermitian metric, and curves in the space of chains with null tangent vectors are shown to correspond to a path of chains tangent to a curve in the boundary transverse to the contact structure. A family of local differential chain curvature operators are introduced which exactly characterize when a transverse curve is a chain. In particular, operators that are invariant under the stabilizer of a point in the interior of complex hyperbolic space, or a point on the boundary, are developed in detail. Finally, these chain curvature operators are used to prove a generalization of Louiville's theorem: a sufficiently smooth mapping from the boundary of complex hyperbolic 2-space to itself which preserves chains must be the restriction of a global automorphism.