A new characterization of hyperellipticity
type: Item
instance of: scholarly article
title: A new characterization of hyperellipticity
zbMATH Open document ID: 0959.30025
author: Bernard Maskit
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: A theorem on the crossing points of closed geodesics was proved by Jørgensen, that is, if \(x\) is the crossing point of two closed geodesics on a surface with a complete metric of constant negative curvature, then there are infinitely many different closed geodesics passing through \(x\) [\textit{T. Jørgensen}, Proc. Am. Math. Soc. 72, 140-142 (1978; Zbl 0406.53011)]. In this paper the author gives another proof of this theorem by using the fact that a fixed point of the conformal involution of a torus with a hole is a crossing point of infinitely many simple closed geodesics. Also, the author gives a necessary and sufficient condition for a closed Riemann surface of genus \(p\geq 2\) to be a hyperelliptic surface by using a geodesic necklace \({\mathcal L}\) on it, where \({\mathcal L}\) is a cyclically ordered set of \(2p+2\) simple nondividing closed geodesics \(L_1,L_2,\ldots , L_{2p+2}\) with the property that each \(L_i\) intersects \(L_{i-1}\) exactly once, intersects \(L_{i+1}\) exactly once, and is disjoint from the other geodesics in \({\mathcal L}\).
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 1548338
zbMATH Keywords: hyperelliptic Riemann surfaces
zbMATH Keywords: torus with a hole
zbMATH Keywords: geodesic necklace
zbMATH Keywords: closed geodesics
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030374665
OpenAlex ID: W1966732514
DOI: 10.1307/MMJ/1030374665
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description: scientific article; zbMATH DE number 1548338