## 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.## References

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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