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

DOI:
https://doi.org/10.1090/S1088-4173-01-00066-2

Published electronically:
February 21, 2001

MathSciNet review:
1836403

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

The trace of a smooth function 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 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

DOI:
https://doi.org/10.1090/S1088-4173-01-00066-2

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