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Conformal Geometry and Dynamics

ISSN 1088-4173



Diffeomorphisms of the circle and hyperbolic curvature

Author: David A. Singer
Journal: Conform. Geom. Dyn. 5 (2001), 1-5
MSC (2000): Primary 53A55; Secondary 52A55
Published electronically: February 21, 2001
MathSciNet review: 1836403
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Abstract | References | Similar Articles | Additional Information


The trace $Tf$ of a smooth function $f$ of a real or complex variable is defined and shown to be invariant under conjugation by Möbius transformations. We associate with a convex curve of class $C^2$ in the unit disk with the Poincaré metric a diffeomorphism of the circle and show that the trace of the diffeomorphism is twice the reciprocal of the geodesic curvature of the curve. Then applying a theorem of Ghys on Schwarzian derivatives we give a new proof of the four-vertex theorem for closed convex curves in the hyperbolic plane.

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Additional Information

David A. Singer
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058

Received by editor(s): July 26, 2000
Received by editor(s) in revised form: January 23, 2001
Published electronically: February 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society