Conformal Geometry and Dynamics

Author(s): David A. Singer
Journal: Conform. Geom. Dyn. 5 (2001), 1-5.
MSC (2000): Primary 53A55; Secondary 52A55
Posted: February 21, 2001
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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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