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Accelerations of Riemannian quadratics

Author: Lyle Noakes
Journal: Proc. Amer. Math. Soc. 127 (1999), 1827-1836
MSC (1991): Primary 53B20, 53B99; Secondary 41A15, 41A29, 41A99
Published electronically: February 18, 1999
MathSciNet review: 1486744
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Abstract: A Riemannian corner-cutting algorithm generalizing a classical construction for quadratics was previously shown by the author to produce a $C^1$ curve $p_\infty$ whose derivative is Lipschitz. The present paper takes the analysis of $p_\infty$ a step further by proving that it possesses left and right accelerations everywhere. Two-sided accelerations are shown to exist on the complement of a countable dense subset $D$ of the domain. The results are shown to be sharp in the following sense. For almost any scaled triple in Euclidean space there is a Riemannian perturbation of the Euclidean metric such that the two-sided accelerations of the resulting curve $p_\infty$ exist nowhere in $D$.

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Additional Information

Lyle Noakes
Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia

Keywords: Geodesic, parallel translation, corner-cutting
Received by editor(s): December 7, 1996
Received by editor(s) in revised form: June 11, 1997
Published electronically: February 18, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society