Generalized Schwarzian derivatives for generalized fractional linear transformations

Abstract

Generalizations of the classical Schwarzian derivative of complex analysis have been proposed by Osgood and Stowe (12, 13), Carne (5), and Ahlfors (3). We present another generalization of the Schwarzian derivative over vector spaces. Introduction. Our approach is to dene an analogue of the Schwarzian derivatives in R(f1g using the Cliord algebra generated from R n . More precisely, we use Vahlen's group of Cliord matrices to construct a \deriva- tive which in appearance bears an extremely close resemblance to the clas- sical Schwarzian derivative. As conformal transformations in dimensions greater than two correspond to Mobius transformations we are forced to introduce a family of Schwarzians in higher dimensions. We show that a C 3 dieomorphism annihilated by this family of Schwarzian derivatives is, up to a linear isomorphism, a Mobius transformation. We also show that these generalized Schwarzian derivatives possess a conformal invariance un- der Mobius transformations, and contain the generalized Schwarzian deriva- tives described by Ahlfors (3). Unfortunately, this work also tells us that the method used for obtaining the chain rule for the classical Schwarzian derivative (see (10)) breaks down in higher dimensions. Motivated by the fact that the analogue of Vahlen's group of Cliord matrices over Minkowski space is U(2; 2) we show that the fractional linear transformations associated withU(2; 2), Sp(n;R), the real symplectic group, andH(n;n), the quaternionic unitary group, all have Schwarzian derivatives associated with them. These transformations have previously been described in (7, 9), and elsewhere. We also show that the conformal group over R p;q has a generalized Schwarzian derivative. Preliminaries. From R n we may construct a Cliord