Generalized Schwarzians in Several Variables and Moebius Invariant DifferentialOperators

摘要

We introduce certain Mobius invariant differential operators in severalvariables, which in many ways correctly generalize the classical Schwarzian derivative in one variable. We show that these generalized Schwarzians generate the entire set of Mobius invariant differential operators acting on local biholomorphisms. We then prove the composition formula for generalized Schwarzians from the point of view of connections in vector bundles. We also give the necessary and sufficient condition to prescribe generalized Schwarzians and give the formula for the solution.