The Schwarzian derivative for maps between manifolds with complex projective connections

On Iterated Positive Schwarzian derivative Maps

The families of one-dimensional maps have been the object of many studies since [Sharkovsky, 1964], for instance in [Sharkovsky et al., 1997] or in [de Melo & van Strien, 1993], whose authors also studied this subject, we have two large surveys of the enormous effort made in the last decades in this field of work. We adopt the usual definitions as for instance symbolic dynamics or topological e...

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 gen...

Stable Exponentially Harmonic Maps Between Finsler Manifolds

The Schwarzian Derivative for Harmonic Mappings

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

Projective Connections and Schwarzian Derivatives for Supermanifolds, and Batalin-vilkovisky Operators

We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction of a Laplacian on the algebra of densities for a projectively connected manif...