Circular geometry and the Schwarzian (Q2706962)

Here the authors give three generalizations of the notion of Schwarzian derivative: the Schwarzian derivative of a curve in a Riemannian manifold, the Schwarzian derivative of a Riemannian metric with respect to another, and the Schwarzian derivative of an immersion between two Riemannian manifolds.NEWLINENEWLINENEWLINEIn the case of a curve \(\gamma\) in a Riemannian manifold \((M,g)\), the Schwarzian derivative \(s^2\gamma\) of \(\gamma\) is a curve in the bundle \(\wedge^0 \text{ T}M \oplus \wedge^2\text{ T}M\). It is shown that if \(\gamma\) is regular, with the \(0\)-part of \(s^2\gamma\) being negative, then \(\gamma\) is injective (Theorem 1.3).NEWLINENEWLINENEWLINETheorem 2.8 relates the Schwarzian derivative of a Riemannian metric with the Schwarzian proposed by \textit{B. Osgood} and \textit{D. Stowe} [Duke Math. J. 67, 57-99 (1992; Zbl 0766.53034)]. Theorem 2.9 shows that, on a manifold of dimension greater than or equal to \(2\), two Riemannian metrics are Möbius equivalent if and only if they are concircularly equivalent.NEWLINENEWLINENEWLINEThe last section contains two theorems (3.3 and 3.8) concerning the Schwarzian derivative of an immersion \(f: M \to N\) between two Riemannian manifolds ; these theorems give sufficient conditions for the injectivity of \(f\). A corollary (3.9) of these results is Nehari's sufficient condition of univalence for local diffeomorphisms of the unit disk.