Diffeomorphisms of the circle and hyperbolic curvature
Author:
David A. Singer
Journal:
Conform. Geom. Dyn. 5 (2001), 15
MSC (2000):
Primary 53A55; Secondary 52A55
Published electronically:
February 21, 2001
MathSciNet review:
1836403
Fulltext PDF Free Access
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Abstract: 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 fourvertex theorem for closed convex curves in the hyperbolic plane.
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 G. Cairns and R.W. Sharpe, On the inversive differential geometry of plane curves, Enseign. Math. (2) 36 (1990), no. 12, 175196. MR 91h:53001
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 C. Duval and V. Ovsienko, Lorentz world lines and Schwarzian derivative, (Russian) Funktsional. Anal. i Prilozhen. 34 (2000), no. 2, 6972; translation in Funct. Anal. Appl. 34 (2000), no. 2, 135137. CMP 2000:17
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 E. Ghys, Cercles osculateurs et géométrie Lorentzienne, Colloquium talk at Journée inaugurale du CMI, Marseille, February 1995.
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 S.B. Jackson, The fourvertex theorem for surfaces of constant curvature, Amer. J. Math. 67 (1945), 563582. MR 7:259h
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 S. Mukhopadhyaya, New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc. 1 (1909), 3137.
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 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, 297307. MR 98g:57050
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 P. Scherk, The fourvertex theorem, Proceedings of the First Canadian Mathematical Conference (Montreal), 1945, Toronto, 1946, pp. 97102. MR 8:485d
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 Ricardo UribeVargas, On the vertex and flattening theorems in higherdimensional Lobatchevskian space, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 505510. MR 2000e:51027
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Additional Information
David A. Singer
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 441067058
Email:
das5@po.cwru.edu
DOI:
http://dx.doi.org/10.1090/S1088417301000662
PII:
S 10884173(01)000662
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
