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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Diffeomorphisms of the circle and hyperbolic curvature
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by David A. Singer PDF
Conform. Geom. Dyn. 5 (2001), 1-5 Request permission

Abstract:

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
  • Email: das5@po.cwru.edu
  • Received by editor(s): July 26, 2000
  • Received by editor(s) in revised form: January 23, 2001
  • Published electronically: February 21, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 5 (2001), 1-5
  • MSC (2000): Primary 53A55; Secondary 52A55
  • DOI: https://doi.org/10.1090/S1088-4173-01-00066-2
  • MathSciNet review: 1836403