A new characterization of hyperellipticity
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 \({\mathcal L}\).
scientific article; zbMATH DE number 1548338