The Schwarzian derivative and Möbius equation on strictly pseudo-convex CR manifolds

The notion of Schwarzian derivative for locally univalent holomorphic functions on complex plane was generalized for conformal diffeomorphisms by Osgood and Stowe in 1992. We shall identify a tensor that may serve as an analogue of the Schwarzian of Osgood and Stowe for CR mappings and then use the tensor to define and study the CR Möbius transformations and metrics of pseudohermitian manifolds. We shall establish basic properties of the CR Schwarzian and a local characterization of the CR spherical manifolds in terms of the fully integrability of the CR Möbius equation. In another direction, we shall prove two rigidity results for the Möbius changes of metrics on compact CR manifolds.

Keywords

CR map, Schwarzian derivative, Möbius transformation

2010 Mathematics Subject Classification

30F45, 32V05

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The author was partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098.

Received 10 December 2015

Published 7 May 2018