and that the Schwarzian is identically zero exactly for Möbius transformations. Let D denote the unit disk. There has been progress in several areas, but the innovations we treat here come primarily from an injectivity criterion for conformal, local diffeomorphisms of an n-dimensional Riemannian manifold into the n-sphere. The criterion involves a generalization of the Schwarzian derivative whi...

We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (R/Z) × I.

The main character of the present note is the Schwarzian derivative, and we start with a brief reminder of its definition and main properties. Let f : RP → RP be a projective line diffeomorphism. For every point x ∈ RP there exists a unique projective transformation gx : RP 1 → RP whose 2-jet at x coincides with that of f . The Schwarzian derivative S(f) measures the deviation of the 3-jet jf f...

By considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal.

We study the Schwarzian derivative from a variational viewpoint. Firstly we show that defines first integral of Euler--Lagrange equation second order Lagrangian. Secondly, itself is operator for an appropriately chosen class variations.