Abstract
In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane.
Highlights
We give a section by section summary of the contents of this paper
We show that K∗ is invariant under the reflection S(j) = Rnj U in the ray
We identify two nonadjacent closed edges of cl(K∗), the closure of K∗, if one edge is obtained from the other space (cl(K∗) \
Summary
We give a section by section summary of the contents of this paper. In §1 we define the Schwarz–Christoffel conformal map FQ (2) of the complex plane less {0, 1} onto a quadrilateral Q, which is formed by reflecting a rational triangle Tn0n1n∞ in the real axis. The quadrilateral Q is holomorphically diffeomorphic to a fundamental domain D of the action of the covering group on Sreg. To construct the model Sreg of the affine Riemann surface Sreg from the regular stellated n-gon K∗ we follow Richens and Berry [2]. Following Richens and Berry [2] we impose the condition: when a geodesic, starting at a point in int(cl(K∗) \ O), meets ∂K∗ it undergoes a reflection in the edge of K∗ that it meets Such geodesics never meet a vertex of cl(K∗). An argument shows that G invariant geodesics on (Sreg, Γ) correspond under the map δK∗\O to billiard motions on (K∗ \ O, γ|(K∗\O)).
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