Affiliation:
1. School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China
2. School of Mathematical Sciences, East China Normal University, Shanghai 200241, P. R. China
Abstract
Rivin interpreted McShane’s identity as an identity for closed geodesics with one self-intersection on a one-cusped hyperbolic torus (cf. [I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012) 2391–2412, Theorem 3.2]). In this note we point out that only those geodesics of non-hyperelliptic type are included in the interpreted identity, while those of hyperelliptic type are missing, and we give the desired identity for the closed geodesics with one self-intersection of hyperelliptic type. We also remark that the interpretation in [I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012) 2391–2412, Theorem 3.7] of generalized McShane’s identity for a one-coned hyperbolic torus with cone angle [Formula: see text] is only partly valid because closed curves with one minimal self-intersection of non-hyperelliptic type are realizable as closed geodesics on the torus if and only if [Formula: see text].
Funder
National Natural Science Foundation of China
Publisher
World Scientific Pub Co Pte Ltd
Subject
Algebra and Number Theory