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JOURNALS // Regular and Chaotic Dynamics // Archive

Regul. Chaotic Dyn., 2005 Volume 10, Issue 4, Pages 463–485 (Mi rcd721)

This article is cited in 4 papers

Bicentennial of C.G. Jacobi

Algebraic closed geodesics on a triaxial ellipsoid

Yu. N. Fedorovab

a Department of Mathematics and Mechanics, Moscow Lomonosov University, Moscow, 119899, Russia
b Department de Matemàtica I, Universitat Politecnica de Catalunya, Barcelona, E-08028 Spain

Abstract: We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in $\mathbb{R}^3$. Such geodesics are either connected components of real parts of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition law for elliptic functions, and the Moser–Trubowitz isomorphism between geodesics on a quadric and finite-gap solutions of the KdV equation. For the case of 3-fold and 4-fold coverings, some explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.

Keywords: closed geodesics, hyperelliptic curves, hyperelliptic coverings.

MSC: 14H52, 37J45, 53C22, 58E10

Received: 13.06.2005
Accepted: 17.10.2005

Language: English

DOI: 10.1070/RD2005v010n04ABEH000326



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