The Schwarzian Derivative and Conformal Mapping of Riemannian Manifolds

S f t,,-,] \-fT,I f 2 t,,--fT,] The Schwarzian is important in many areas of complex analysis (see, for example, the recent book of O. Lehto ILl) but it occurs first and foremost through its connection with M6bius transformations. The basic facts are az+b (1.1) S(f) 0 if and only iff(z) ad bc :/: 0 cz+d’ and (1.2) S(fo h) S(h) if and only iff is M6bius. Equation (1.2) is a special case of a general formula for the Schwarzian of the composite of two analytic functions, which reads (1.3) S(fo h)= S(h) + (S(f)o h)(h’) A generalization of (1.3) will be important for our work. Let (M, g) be a Riemannian manifold of dimension n > 2 and let V denote the Riemannian connection for the metric g ( ). For a smooth function 40: M ---> R we define a tensor