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

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Abstract | References | Similar Articles | Additional Information

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.

- G. Cairns and R. W. Sharpe,
*On the inversive differential geometry of plane curves*, Enseign. Math. (2)**36**(1990), no. 1-2, 175–196. MR**1071419**
DO C. Duval and V. Ovsienko, - V. Ovsienko and S. Tabachnikov,
*Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves*, Selecta Math. (N.S.)**2**(1996), no. 2, 297–307. MR**1414890**, DOI https://doi.org/10.1007/BF01587937
Sc P. Scherk, - Ricardo Uribe Vargas,
*On the $(2k+2)$-vertex and $(2k+2)$-flattening theorems in higher-dimensional Lobatchevskian space*, C. R. Acad. Sci. Paris Sér. I Math.**325**(1997), no. 5, 505–510 (English, with English and French summaries). MR**1692315**, DOI https://doi.org/10.1016/S0764-4442%2897%2988897-0

*Lorentz world lines and Schwarzian derivative*, (Russian) Funktsional. Anal. i Prilozhen.

**34**(2000), no. 2, 69–72; translation in Funct. Anal. Appl.

**34**(2000), no. 2, 135–137. Gh E. Ghys,

*Cercles osculateurs et géométrie Lorentzienne*, Colloquium talk at Journée inaugurale du CMI, Marseille, February 1995. Ja S.B. Jackson,

*The four-vertex theorem for surfaces of constant curvature*, Amer. J. Math.

**67**(1945), 563–582. Mu S. Mukhopadhyaya,

*New methods in the geometry of a plane arc*, Bull. Calcutta Math. Soc.

**1**(1909), 31–37.

*The four-vertex theorem*, Proceedings of the First Canadian Mathematical Conference (Montreal), 1945, Toronto, 1946, pp. 97–102.

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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

Article copyright:
© Copyright 2001
American Mathematical Society