The Schwarzian derivative, conformal connections, and Möbius structures (Q1283057)

The Schwarzian derivative in complex analysis measures to what extent a conformal transformation differs from being a Möbius transformation, i.e. from mapping circles to circles. The purpose of this paper is to define the notion of Schwarzian derivative for conformal transformations between manifolds of arbitrary dimension. The extra structure on a conformal manifold needed for the definition of the Schwarzian derivative to make sense is a Möbius structure, which is a special kind of conformal connection. After discussing the classical Schwarzian derivative, the authors give a detailed review of the relevant facts on conformal structures and connections. They define Möbius structures and Schwarzian derivatives and explain, how a Riemannian metric gives rise to a unique Möbius structure. In the last section it is proved that -- as in the classical theory -- the Schwarzian derivative measures the failure of a conformal transformation to map circles to circles.