Journal
JOURNAL OF GEOMETRY AND PHYSICS
Volume 61, Issue 5, Pages 899-921Publisher
ELSEVIER
DOI: 10.1016/j.geomphys.2011.01.001
Keywords
Hyperelliptic integrals and their inversion; theta functions; theta constants; Geodesic equation
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Funding
- German Research Foundation DFG
- Hanse-Wissenschaftskolleg (Institute for Advanced Study) in Delmenhorst
- center of excellence QUEST
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The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is defined only locally, and is clone using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the theta-divisor. For a representation of the hyperelliptic functions the Klein-Weierstra ss multivariable sigma function is introduced. It is shown that all parameters needed for the calculations like period matrices and Abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and theta-constants. The cases of genus 2 and genus 3 are considered in detail. The method is exemplified by particle motion associated with a genus 3 hyperelliptic curve. (c) 2011 Elsevier B.V. All rights reserved.
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