An affine model of a Riemann surface associated to a Schwarz-Christoffel mapping

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.

Because F T | [0,1] is real valued, we may use the Schwarz reflection principle to extend F T to the holomorphic diffeomorphism Here Q = Q n0n1n∞ is a quadrilateral with internal angles 2π n0 n , π n∞ n , 2π n1 n , and π n∞ n and vertices at O, D, C, and D, see figure 2. The conformal mapping F Q 2 An associated affine Riemann surface S Let ξ and η be coordinate functions on C 2 . Consider the affine Riemann surface S = S n0,n1,n∞ in C 2 , associated 3 to the holomorphic mapping F Q , defined by g(ξ, η) = η n − ξ n−n0 (1 − ξ) n−n1 = 0.
Below is a table listing all the partitions {n 1 , n 0 , n ∞ } of n, which give a low genus Riemann surface S = S n0,n1,n∞ g n 0 , n 1 , n ∞ ; n g n 0 , n 1 , n ∞ ; n

Corollary 2.1a
If n is an odd prime number and {n 1 , n 0 , n ∞ } is a partition of n into three parts, then the genus of S is 1 2 (n − 1). Proof. Because n is prime, we get d 0 = d 1 = d ∞ = 1. Using (9) we obtain g = 1 2 (n − 1).
Proof. We only need to show that π is a covering map. First we note that every fiber of π is a finite set with n elements, since for each fixed ξ ∈ C \ {0, 1} we have π −1 (ξ) = {(ξ, η) ∈ S reg η = ω k (ξ n−n0 (1 − ξ) n−n1 ) 1/n }. Here ω k for k = 0, 1, . . . , n − 1, is an n th root of 1 and ( ) 1/n is the complex n th root used in the definition of the Schwarz-Christoffel map F Q (2). Hence the map π is a proper surjective holomorphic submersion, because each fiber is compact. Thus the mapping π is a presentation of a locally trivial fiber bundle with fiber consisting of n distinct points. In other words, the map π is a n to 1 covering mapping.
Clearly R n = id C 2 = e, the identity element of G and G = {e, R, . . . , R n−1 }.
Since G is finite, and hence is compact, the action Φ is proper. For every g ∈ G we have Φ g (0, 0) = (0, 0) and Φ g (1, 0) = (1, 0). So Φ g maps S reg into itself. At (ξ, η) ∈ S reg the isotropy group G (ξ,η) is {e}, that is, the G-action Φ on S reg is free. Thus the orbit space S reg / G is a complex manifold.
Thus Φ R is a covering transformation for the bundle presented by the mapping π. So G is a subgroup of the group of covering transformations. These groups are equal because G acts transitively on each fiber of the mapping π.

Another model for S reg
In this section we construct another model S reg for the smooth part S reg of the Riemann surface S (3).
Let D ⊆ S reg be a fundamental domain for the G action Φ (14) on S reg . So (ξ, η) ∈ D if and only if for ξ ∈ C \ {0, 1} we have η = ξ n−n0 (1 − ξ) n−n1 1/n . Here ( ) 1/n is the n th root used in the definition of the mapping F Q (2). The domain D is a connected subset of S reg with nonempty interior. Its image under the map π (11) is C \ {0, 1}. Thus D is one "sheet" of the covering map π. So π| D is one to one. We now give a more geometric description of the fundamental domain D. Consider the mapping where the map π is given by equation (11). The map δ is a holomorphic diffeomorphism of int D onto int Q, which sends ∂D homeomorphically onto ∂Q. Look  Here R is the rotation C → C : z → e 2πi/n z. We say that the quadrilateral Q = Q 2n0,n∞,2n1,n∞ forms K * less its vertices, see figure 4 above.
Claim 3.1 The connected set K * is a regular stellated n-gon with its 2n vertices omitted, which is formed from the quadrilateral Q = OD CD , see figure 5.
Proof. By construction the quadrilateral Q = OD CD is contained in the So K * = n j=0 R j (Q ). Thus K * is the regular stellated n-gon, one of whose sides is the diagonal D D of Q . We would like to extend the mapping δ (16) to a mapping of S reg onto K * . Let where Φ is the G action defined in equation (14). So we have a mapping , since D is a fundamental domain for the Gaction Φ (14) on S reg . Because K * = 0≤j≤n−1 R j δ(D) , the mapping δ K * is surjective. Hence δ K * is holomorphic, since it is continuous on S reg and is holomorphic on the dense open subset 0≤j≤n−1 R j (int D) of S reg .
Let U : C → C : z → z and let G be the group generated by the rotation R and the reflection U subject to the relations R n = U 2 = e and RU = U R −1 , or The closure cl(K * ) of K * = 0≤j≤n−1 R j (Q) is invariant under G and hence under the rotation R. Because the quadrilateral Q is invariant under the reflection U : z → z, and U R j = R −j U , it follows that cl(K * ) is invariant under the reflection U . Thus cl(K * ) is invariant under the group G. We now look at some group theoretic properties of K * . Lemma 3.2 If F is a closed edge of the polygon cl(K * ) and g| F = id| F for some g ∈ G, then g = e.
Proof. Suppose that g = e. Then g = R p U for some ∈ {0, 1} and some p ∈ {0, 1, . . . , n − 1}. Let g = R p U and suppose that F is an edge of cl( is a reflection in the closed ray j = {te i πnj /n ∈ C t ∈ OD}. The closed ray 0 is the closure of the side OD of the quadrilateral Q = ODCD in figure 5. Proof. S (j) fixes every point on the closed ray j , because Since (S (j) ) 2 = (R nj U )(R nj U ) = R nj (U U )R −nj = e, it follows that S (j) is a reflection in the closed ray j .
k is a reflection in the closed ray R k j .
Proof. This follows because (S (j) k ) 2 = R k (S (j) ) 2 R −k = e and S (j) k fixes every point on the closed ray R k j , for Corollary 3.3b For every j = 0, 1, ∞, every S (j) k with k = 0, 1, . . . , n − 1, and every g ∈ G, we have gS Proof. We compute. For every k = 0, 1, . . . , n − 1 we have and U S (j) where t = −(k +2n j ) mod n. Since R and U generate the group G, the corollary follows.
Corollary 3.3c For j = 0, 1, ∞ let G j be the group generated by the reflections S (j) k for k = 0, 1, . . . , n − 1. Then G j is a normal subgroup of G. Proof. Clearly G j is a subgroup of G. From equations (18) and (19) it follows that gS (j) k g −1 ∈ G j for every g ∈ G and every k = 0, 1 . . . , n − 1, since G is generated by R and U . But G j is generated by the reflections S (j) k for k = 0, 1, . . . , n − 1, that is, every g ∈ G j may be written as S As a first step in constructing S reg from the regular stellated n-gon K * we look at certain pairs of edges of cl(K * ). We say two distinct closed edges E and E of cl(K * ) are adjacent if and only if they intersect at a vertex of cl(K * ). For Geometrically, two nonadjacent closed edges E and E of cl(K * ) are equivalent if and only if E is obtained from E by reflection in the line R m j for some m ∈ {0, 1, . . . , n − 1}.
In figure 7, where K * = K * 1,1,4 , parallel edges of K * , which are labeled with the same letter, are G 0 -equivalent. This is no coincidence.
Lemma 3.4 Let K * be formed from the quadrilateral Q = T ∪T , where T is the isosceles triangle T n0n0n∞ less its vertices. Then nonadjacent edges of ∂ cl(K * ) are G 0 -equivalent if and only if they are parallel.
Proof. In figure 6 let OAB be the triangle T with ∠AOB = α, ∠OAB = β, and ∠ABO = γ. Let OABA be the quadrilateral formed by reflecting the triangle OAB in its edge OB. The quadrilateral OABA reflected it its edge OA is the quadrilateral OAB A . Let AC be perpendicular to A B and AC be perpendicular to A B, see figure 6. Then CAC is a straight line if and only if ∠C AB +∠B AB +∠BAC = π. By construction ∠C AB = ∠BAC = π/2−2γ and ∠B AB = 2π − 2β. So if and only if α = γ. Hence the edges A B and A B are parallel if and only if the triangle OAB is isosceles. We now study some group theoretic properties of the set of equivalent edges of cl(K * ), which will be used when determining the topology of S reg . Let E be the set of unordered pairs [E, E ] of nonadjacent edges of cl(K * ). Define an for every unordered pair [E, E ] of nonadjacent edges of cl(K * ). For every g ∈ G the edges g(E) and g(E ) are nonadjacent. This follows because the edges E and E are nonadjacent and the elements of G are invertible mappings of C into itself.
. Thus the mapping • is well defined. It is an action because for every g and h ∈ G we have The action • of G on E induces an action · of the group G j of reflections on the set E j of equivalent edges of cl(K * ), which is defined by for every g j ∈ G j , every edge E of cl(K * ), and every generator S because G j is a normal subgroup of G by corollary 3.3c. Since and gS (j) if and only if if and only if one of the statements 1) To see this we argue as follows. If g = e, then g = R p (S (j) ) for some = 0, 1 and some p ∈ {0, 1, . . . , n − 1}, see equation (20). Suppose that g = R p with p = 0. Then g(E) = E, which contradicts our hypothesis.
We now begin the construction of S reg by identifying equivalent edges of cl(K * ). Let [E, S (0) m (E)] be an unordered pair of equivalent closed edges of cl(K * ). We say that x and y in cl(K * ) are equivalent, x ∼ y, if 1) x and y lie in ∂ cl(K * ) with x ∈ E and y = S 1} or 2) x and y lie in int cl(K * ) and x = y. The relation ∼ is an equivalence relation on cl(K * ). Let cl(K * ) ∼ be the set of equivalence classes and let be the map which sends p to the equivalence class [p], that contains p. 6 Give cl(K * ) the topology induced from C and put the quotient topology on cl(K * ) ∼ .
Proof. To show that (K * \{O}) ∼ is a smooth manifold, let E + be an open edge of K * . For p + ∈ E + let D p+ be a disk in C with center at p + , which does not contain a vertex of cl(K * ). Set D + which is a smooth 2-disk, since the identification mapping π is the identity on int K * . It follows that (K * \ {O}) ∼ is a smooth 2-dimensional manifold without boundary.
We now handle the vertices of cl(K * ). Let v + be a vertex of cl(K * ) and set with r ≥ 0 and 0 ≤ θ ≤ 1 is a homeomorphism, which sends the wedge with angle π to the wedge with angle πs. The latter wedge is formed by the closed edges E + and E + of cl(K * ), which are adjacent at the vertex v + such that e iπs E + = E + with the edge E + being swept out through int cl(K * ) during its rotation to the edge E + . Because cl(K * ) is a rational regular stellated n-gon, the value of s is a rational number for each vertex of cl( We now describe a triangulation of K * \ {O}. Let T = T 1,n1,n−(1+n1) be the open rational triangle OCD with vertex at the origin O, longest side OC on the real axis, and interior angles 1 n π, n1 n π, and n−1−n1 n π. Let Q be the quadrilateral T ∪ T . Then Q is a subset of the quadrilateral Q = ODCD, see figure 5. Moreover K * = n−1 =0 R (Q ). The 2n triangles cl(R j (T )) \ {O} and cl(R j (T )) \ {O} with j = 0, 1, . . . , n − 1 form a triangulation T K * \{O} of K * \ {O} with 2n vertices R j (C) and R j (D ) for j = 0, 1, . . . , n − 1; 4n open edges R j (OC), R j (OD ), R j (CD ), and R j (CD ) for j = 0, 1, . . . , n − 1; and 2n open triangles The action of G on cl(K * ) preserves the set of unordered pairs of equivalent edges of cl(K * ) in E j for j = 0, 1, ∞. Hence G induces an action on cl(K * ) ∼ , which is proper, since G is finite. The G action is free on K * \ {O} and thus on (K * \ {O}) ∼ by lemma 3.6. We have proved We now determine the topology of the orbit space S reg . For j = 0, 1, ∞ and = 0, 1, . . . , d j − 1 let A j be an end point of a closed edge E of cl(K * ), which lies on the unordered pair [E, )), where j = 0, 1, ∞ and = 0, 1, . . . , d j −1; 3n open edges µ(R j (OC)), µ(R j (OD )), and µ(R j (CD)) for j = 0, 1, . . . , n − 1; and 2n open triangles µ(R j (T )) and µ(R j (T )) for j = 0, 1, . . . n − 1. Thus the Euler characteristic Theorem 3.11 Let K * be the regular stellated n-gon formed from the rational quadrilateral Q n0n1n∞ with d j = gcd(n j , n) for j = 0, 1, ∞. The G orbit space S reg formed by first identifiying equivalent edges of the regular stellated n-gon K * less O and then acting on the identification space by the group G, is a smooth 2-sphere with g handles, where 2g = n + 2 − (d 0 + d 1 + d ∞ ) less some points corresponding to the image of the vertices of cl(K * ) under the map µ.
Since the quadrilateral Q is a fundamental domain for the action of G on K * , the G orbit map µ = σ• π : K * ⊆ C → S restricted to Q is a bijective continuous open mapping. But δ Q : D ⊆ S reg → Q ⊆ C is a bijectives continous open mapping of the fundamental domain D of the G action on S. Consequently, the G orbit space S is homeomorphic to the G orbit space S. The mapping mu is holomorphic except possibly at 0 and the vertices of cl(K * ). So the mapping µ• δ K * : S reg → S reg is a holomorphic diffeomorphism.
Example 3.12 The following example illustrates theorem 3.11 when K * = K * 1,1,4 is a regular stellated hexagon formed by repeatedly rotating the quadrilateral Q = OD CD by R through an angle 2π/6, see figure 7.
Let G 0 be the group generated by the reflections S Here S (0) = RU is the reflection which leaves the closed ray 0 = {te iπ/6 t ∈ OD } fixed. Define an equivalence relation on cl(K * ) be the identification map which sends a point p ∈ cl(K * ) to the equivalence class [p], which contains p. Give cl(K * ) the topology induced from C. Placing the quotient topology on cl(K * ) ∼ turns it into a connected compact topological manifold without boundary. Let K * be cl(K * ) less its vertices. The identification space (K * \ {O}) ∼ = π(K * \ {O}) is a connected compact 2-dimensional smooth manifold without boundary.
preserves equivalent edges of cl(K * ) and is free on K * \ {O}. Hence it induces a G action on (K * \ {O}) ∼ , which is free and proper. Thus its orbit map is surjective, smooth, and open. The orbit space S reg = σ((K * \ {O}) ∼ ) is a connected compact 2-dimensional smooth manifold. The identification space (K * \ {O}) ∼ has the orientation induced from an orientation of K * \ {O}, which comes from C. So S reg has a complex structure, since each element of G is a conformal mapping of C into itself. Our aim is to specify the topology of S reg .
Consider the induced action of G on the set E 0 of unordered pairs of equivalent closed edges of cl(K * ), that is, E 0 is the set [E, S k (E)] for k = 0, 1, . . . , 5, where E is a closed edge of cl(K * ). Table 1 lists the elements of E 0 .  G acts on E 0 , namely, g ·[E, S Since G 0 is the group generated by the reflections S (0) k , k = 0, 1, . . . , 5, it is a normal subgroup of G. Hence the action of G on E 0 restricts to an action of G 0 on E 0 and permutes G 0 -orbits in E 0 . Thus the set of G 0 -orbits in E 0 is G-invariant. We now look at the G 0 -orbits on E 0 . We compute the G 0 -orbit of d ∈ E 0 as follows. We have Suppose that B is an end point of the closed edge E of cl(K * ). Then E lies in a unique [E, S    To determine the topology of the G orbit space S reg we find a triangulation of S reg . Note that the triangulation T K * \{O} of K * \ {O}, illustrated in figure  7, is G-invariant. Its image under the identification map π is a G-invariant triangulation T (K * \{O} of (K * \ {O}) ∼ . After identification of equivalent edges, each vertex π(v), each open edge π(E), having π(O) as an end point, or each open edge π([F, F ]), where [F, F ] is a pair of equivalent edges of cl(K * ), and each open triangle π(T ) in T (K * \{O}) ∼ lies in a unique G orbit. It follows that σ(π(v)), σ(π(E)) or σ(π([F, F ])), and σ(π(T )) is a vertex, an open edge, and an open triangle, respectively, of a triangulation T Sreg = σ(T (K * \{O}) ∼ ) of S reg . The triangulation T Seg has 4 vertices, corresponding to the G orbits σ(π(O(D ))), σ(π(O(D ))), σ(π(O(C))), and σ(π(O(D &D ))); 18 open edges corresponding to σ(π(R j (OC))), σ(π(R j (OD ))), and σ(π(R j (CD ))) for j = 0, 1, . . . , 5; and 12 open triangles σ(π(R j ( OCD ))) and σ(π(R j ( OCD ))) for j = 0, 1, . . . , 5. Thus the Euler characteristic χ(S reg ) of S reg is 4 − 18 + 12 = −2. Since S reg is a 2-dimensional smooth real manifold, χ(S reg ) = 2 − 2g, where g is the genus of S reg . Hence g = 2. So S reg is a smooth 2-sphere with 2 handles, less a finite number of points, which lies in a compact topological space S = cl(K * ) ∼ /G, that is its closure.

An affine model of S reg
We construct an affine model of the Riemann surface S reg .
We return to the stellated regular n-gon K * = K * n0n1n∞ , which is formed from the quadrilateral Q = Q n0n1n∞ less its vertices. Repeatedly reflecting in the edges of K * and then in the edges of the resulting reflections of K * et cetera, we obtain a covering of C \ V + by certain translations of K * . Here V + is the union of the translates of the vertices of K * and its center O. Let T be the group generated by these translations. The semidirect product G = G T acts freely and properly on C\V + . It preserves equivalent edges of C\V + and it acts freely and properly on (C \ V + ) ∼ , the space formed by identifying equivalent edges in C \ V + . The orbit space (C \ V + ) ∼ /G is holomorphically diffeomorphic to S reg and is the desired affine model. We now justify these assertions.
Proof. From figure 9 we have ∠D CO = n1 n π. So ∠COH = 1 2 π − n1 n π. Hence the line 0 , containing the edge CD of K * , is perpendicular to the direction e [ 1 2 − n 1 n ]π . Since COD is the reflection of COD in the line segment OC, the line 1 , containing the edge CD of K * , is perpendicular to the direction e [− 1 2 + n 1 n ]π . Because the stellated regular n-gon K * is formed by repeatedly rotating the quadrilateral Q = OD CD through an angle 2π n , we find that equation (24) holds. Figure 9. The regular stellated n-gon K ⇤ two of whose sides are CD 0 and CD 0 .
Since ∠COH = 1 2 π − n1 n π, it follows that |H| = |C| sin π n1 n is the distance from the center O of K * to the line 0 containing the side CD , or to the line 1 containing the side CD . So u 0 = (|C| sin π n1 n )e Since the stellated regular n-gon K * is formed by repeatedly rotating the quadrilateral Q = OD CD through an angle 2π n , the point lies on the line 2j = R j 0 , which contains the edge R j (CD ) of K * ; while lies on the line 2j+1 = R j 1 , which contains the edge R j (CD ) of K * for every j ∈ {0, 1, . . . , n − 1}. Also the line segments Ou 2j and Ou 2j+1 are perpendicular to the line 2j and 2j+1 , respectively, for j ∈ {0, 1, . . . , n − 1}.
The set V of vertices of the stellated regular n-gon K * is is the collection of vertices and centers of the congruent regular stellated n-gons K * , K * k1 , K * k0k1 , . . .. Proof. This follows immediately from lemma 4.2.

Corollary 4.2b
The union of K * , K * k0 , K * k0k1 , . . . K * k0k1···k , . . ., where ≥ 0, 0 ≤ j ≤ , and 0 ≤ k j ≤ 2n − 1, covers C \ V + , that is, Proof. This follows immediately from K * k0k1···k = τ k • · · · • τ k0 (K * ). Let T be the abelian subgroup of the 2-dimensional Eulcidean group E(2) generated by the translations τ j (30) for j = 0, 1, . . . 2n − 1. It follows from corollary 4.2b that the regular stellated n-gon K * with its vertices and center removed is the fundamental domain for the action of the abelian group T on C\V + . The group T is isomorphic to the abelian subgroup T of (C, +) generated by {2u j } 2n−1 j=0 . Consider the group G = G T ⊆ G × T, which is the semidirect product of the dihedral group G, generated by the rotation R through 2π/n and the reflection U subject to the relations R n = e = U 2 and RU = U R −1 , and the abelian group T. An element (R j U , 2u k ) of G is the affine linear map which is the composition of the affine linear map (R j U , 2u k ) followed by (R j U , 2u k ). The mappings G → G : R j → (R j U , 0) and T → G : 2u k → (e, 2u k ) are injective, which allows us to identify the groups G and T with their image in G. Using (32) we may write an element (R j U , 2u k ) of G as (e, 2u k ) · (R j U , 0). So Hence (e, 2u (j+2k) mod 2n ) · (R k U , 0) = (R k U , 0) · (e, 2u j ), which is just equation (29). The group G acts on C as E(2) does, namely, by affine linear orthogonal mappings. Denote this action by ψ : G × C → C : ((g, τ ), z) → τ (g(z)).

Lemma 4.5
The action of T (and hence G) on C \ V + is transitive.
Proof. Let K * k0···k and K * k 0 ···k lie in Since K * k0···k = τ k • · · · • τ k0 (K * ) and K * The action of G on C \ V + is proper because G is a discrete subgroup of E(2) with no accumulation points.
. Then E j is the set of unordered pairs of equivalent edges of C \ V + . Define an action of G on E j by Define a relation ∼ on C \ V + as follows. We say that x and y Then ∼ is an equivalence relation on C \ V + . Let (C \ V + ) ∼ be the set of equivalence classes and let Π be the map which assigns to every p ∈ C \ V + the equivalence class [p] containing p.
Proof. This follows immediately from the definition of the maps Π and π.
Lemma 4.7 The usual action of G on C, restricted to C \ V + , is compatible with the equivalence relation ∼, that is, if x, y ∈ C \ V and x ∼ y, then (g, τ )(x) ∼ (g, τ )(y) for every (g, τ ) ∈ G.
Proof. Suppose that x ∈ F = τ (E), where τ ∈ T . Then y ∈ F = τ (E ), since x ∼ y. So for some S Because of lemma 4.7, the usual G-action on C \ V + induces an action of G on (C \ V + ) ∼ .

Lemma 4.8
The action of G on (C \ V + ) ∼ is free and proper.
Proof. The following argument shows that it is free. Using lemma 3.6 we see that an element of G, which lies in the isotropy group G [F,F ] for [F, F ] ∈ E 0 , interchanges the edge F with the equivalent edge F and thus fixes the equivalence class [p] for every p ∈ F . Hence the G action on (C \ V + ) ∼ is free. It is proper because G is a discrete subgroup of the Euclidean group E (2).
Proof. This claim follows from the fact that the fundamental domain of the Gaction on C \ V + is K * \ {O}, which is the fundamental domain of the G-action on K * \ {O}. Thus Π(C \ V + ) is a fundamental domain of the G-action on (C \ V + ) ∼ , which is equal to π(K * \ {O}) = (K * \ {O}) ∼ by lemma 4.6. Hence the G-orbit space (C\V + ) ∼ /G is equal to the G-orbit space S reg . So the identity map from Π(C \ V + ) to (K * \ {O}) ∼ induces a holomorphic diffeomorphism of orbit spaces.
Because the group G is a discrete subgroup of the 2-dimensional Euclidean group E(2), the compact Riemann surface (C \ V + ) ∼ /G is an affine model of the affine Riemann surface S reg .

The developing map and geodesics
In this section we show that the mapping straightens the holomorphic vector field X (12) on the fundamental domain D ⊆ S reg . We verify that X is the geodesic vector field for a flat Riemannian metric Γ on D.
First we rewrite equation (13) as From the definition of the mapping F Q (2) we get where we use the same complex n th root as in the definition of F Q . This implies For each (ξ, η) ∈ D using (37) and (38) we get So the holomorphic vector field X (12) on D and the holomorphic vector field ∂ ∂z on Q are δ-related. Hence δ sends an integral curve of the vector field X starting at (ξ, η) ∈ D onto an integral curve of the vector field ∂ ∂z starting at z = δ(ξ, η) ∈ Q. Since an integral curve of ∂ ∂z is a horizontal line segment in Q, we have proved Claim 5.1 The holomorphic mapping δ (36) straightens the holomorphic vector field X (12) on the fundamental domain D ⊆ S reg .
We can say more. Let u = Re z and v = Im z. Then is the flat Euclidean metric on C. Its restriction γ| C\V + to C \ V + is invariant under the group G, which is a subgroup of the Euclidean group E(2).
Consider the flat Riemannian metric γ| Q on Q, where γ is the metric (39) on C. Pulling back γ| Q by the mapping F Q (2) gives a metric Pulling the metric γ back by the projection mapping π : Restricting Γ to the affine Riemann surface S reg gives Γ = 1 η dξ 1 η dξ. Lemma 5.2 Γ is a flat Riemannian metric on S reg .
Because D has nonempty interior and the map δ (36) is holomorphic, it can be analytically continued to the map since δ = δ Q | D . By construction δ * Q (γ| Q ) = Γ. So the mapping δ Q is an isometry of (S reg , Γ) onto (Q, γ| Q ). In particular, the map δ is an isometry of (D, Γ| D ) onto (Q, γ| Q ). Moreover, δ is a local holomorphic diffeomorphism, because for every (ξ, η) ∈ D, the complex linear mapping T (ξ,η) δ is an isomorphism, since it sends X(ξ, η) to ∂ ∂z z=δ (ξ,η) . Thus δ is a developing map in the sense of differential geometry. 8 The map δ is local because the integral curves of ∂ ∂z on Q are only defined for a finite time, since they are horizontal line segments in Q. Thus the integral curves of X (12) on D are defined for a finite time. Since the integral curves of ∂ ∂z are geodesics on (Q, γ| Q ), the image of a local integral curve of ∂ ∂z under the local inverse of the mapping δ is a local integral curve of X. This latter local integral curve is a geodesic on (D, Γ| D ), since δ is an isometry. Thus we have proved Claim 5.3 The holomorphic vector field X (12) on the fundamental domain D is the geodesic vector field for the flat Riemannian metric Γ| D on D.
Corollary 5.3a The holomorphic vector field X on the affine Riemann surface S reg is the geodesic vector field for the flat Riemannian metric Γ on S reg .
Proof. The corollary follows by analytic continuation from the conclusion of claim 5.3, since int D is a nonempty open subset of S reg and both the vector field X and the Riemannian metric Γ are holomorphic on S reg .
So the additional relation UR −1 = R U holds. Thus G is isomorphic to the dihedral group D 2n . Lemma 6.1 G is a group of isometries of (S reg , Γ).
Lemma 6.2 G is a group of isometries of (C, γ).
Proof. This follows because R and U are Euclidean motions.
We would like the developing map δ Q (40) to intertwine the actions of G and G and the geodesic flows on (S reg , Γ) and (Q, γ| Q ). There are several difficulties. The first is: the group G does not preserve the quadrilateral Q. To overcome this difficulty we extend the mapping δ Q (40) to the mapping δ K * (17) of the affine Riemann surface S reg onto the regular stellated n-gon K * . Lemma 6.3 The mapping δ K * (17) intertwines the action Φ (14) of G on S reg with the action Ψ : of G on the regular stellated n-gon K * .
Proof. From the definition of the mapping δ K * we see that for each (ξ, η) ∈ D we have δ K * R j (ξ, η) = R j δ K * (ξ, η) for every j ∈ Z. By analytic continuation we see that the preceding equation holds for every (ξ, η) ∈ S reg . Since F Q (ξ) = F Q (ξ) by construction and π(ξ, η) = ξ (11), from the definition of the mapping δ (36) we get δ(ξ, η) = δ(ξ, η) for every (ξ, η) ∈ D. In other words, δ K * U(ξ, η) = U δ K * (ξ, η) for every (ξ, η) ∈ D. By analytic continuation we see that the preceding equation holds for all (ξ, η) ∈ S reg . Hence on S reg we have The mapping ϕ : G → G sends the generators R and U of the group G to the generators R and U of the group G, respectively. So it is an isomorphism.
There is a second more serious difficulty: the integral curves of ∂ ∂z run off the quadrilateral Q in finite time. We fix this by requiring that when an integral curve reaches a point P on the boundary ∂Q of Q, which is not a vertex, it undergoes a specular reflection at P . 9 To make this precise, we give Q the orientation induced from C and suppose that the incoming (and hence outgoing) straight line motion has the same orientation as ∂Q. If the incoming motion makes an angle α with respect to the inward pointing normal N to ∂Q at P , then the outgoing motion makes an angle α with the normal N . 10 Specifically, if the incoming motion to P is an integral curve of ∂ ∂z , then the outgoing motion, after reflection at P , is an integral curve of R −1 ∂ ∂z = e −2πi/n ∂ ∂z . Thus the outward motion makes a turn of −2π/n at P towards the interior of Q, see figure 10 (left). In figure 10 (right) the incoming motion has the opposite orientation from ∂Q. This extended motion on Q is called a billiard motion. A billiard motion starting in the interior of cl(Q) is defined for all time and remains in cl(Q) less its vertices, since each of the segments of the billiard motion is a straight line parallel to an edge of cl(Q) and does not hit a vertex of cl(Q), see figure 12.
We can do more. If we apply a reflection S in the edge of Q in its boundary ∂Q, which contains the reflection point P , to the initial reflected motion at P , and then again to the extended straight line motion in S(Q) when it reaches ∂S(Q), et cetera, we see that the extended motion becomes a billiard motion in the stellated regular n-gon K * = Q ∪ 0≤k≤n−1 SR k (Q) , see figure 12. Figure 11. A periodic billiard motion in the equilateral triangle T = T 1,1,1 starting at P. First, extended by the reflection U to a periodic billiard motion in the quadrilateral Q = T [U(T ). Second, extended by the relection S to a periodic billiard motion in Q [ S(Q). Third, extended by the reflection SR to a periodic billiard motion in the stellated equilateral triangle

So we have verified
Claim 6.4 A billiard motion in the stellated regular n-gon K * , which starts at a point in the interior of K * \ {0}, that does not hit a vertex of cl(K * ), is invariant under the action of the isometry subgroup G of the isometry group G of (K * , γ| K * ) generated by the rotation R.
We now show Lemma 6.5 The holomorphic vector field X (12) on S reg is G-invariant.
Proof. We compute. For every (ξ, η) ∈ S reg and for R ∈ G we have Hence for every j ∈ Z we get for every (ξ, η) ∈ S reg . In other words, the vector field X is invariant under the action of G on S reg . Corollary 6.5a For every (ξ, η) ∈ D we have Proof. Equation (44) is a rewrite of equation (43).
Proof. This follows immediately from the lemma.
We now show Theorem 6.7 The image of a geodesic on (S reg , Γ) under the developing map δ K * (17) is a billiard motion in K * .
Proof. Because Φ R j and Ψ R j are isometries of (S reg , Γ) and (K * , γ| K * ), respectively, it follows from equation (42) that the surjective map δ K * : (S reg , Γ) → (K * , γ| K * ) (17) is an isometry. Hence δ K * is a developing map. Using the local inverse of δ K * and equation (45), it follows that a billiard motion in int(K * \{0}) is mapped onto a geodesic in (S reg , Γ), which is possibly broken at the points (ξ i , η i ) = δ −1 K * (p i ). Here p i ∈ ∂K * are the points where the billiard motion undergoes a reflection. But the geodesic on S reg is smooth at (ξ i , η i ) since the geodesic vector field X is holomorphic on S reg . Thus the image of the geodesic under the developing map δ K * is a billiard motion.
Next we follow a G-invariant set of billiard motions in (K * , γ| K * ), which is the union of an R-invariant billiard motion and its U reflection. After identification of equivalent edges of cl(K * ), see figure 13 (left) and (center), we get a motion on the Riemann surface S reg , which is a geodesic for the induced Riemannian metric γ on the G-orbit space (C \ V + ) ∼ /G, see figure 13 (right). We now justify these assertions.
A billiard motion γ z in the regular stellated n-gon K * , which starts at a point z in the interior of cl(K * )\{O} and does not hit a vertex of cl(K * ), is made up of line segments, each of which is parallel to an edge of cl(K * ). It is invariant under the subgroup G of G generated by the rotation R. Let Rfl γz = {p ∈ ∂ cl(K * ) p = γ z (T p ) for some T p ∈ R} be the set of reflection points in the boundary of cl(K * ) of the billiard motion γ z . Since γ z is invariant under the group G, the set Rfl γz of reflection points is invariant under group G. Because γ z does not hit a vertex of cl(K * ), z is not fixed by the reflection U . The billiard motion γ z starting at z = U (z) is invariant under the group G and, by uniqueness of billiard motions with a given starting point, is equal to the billiard motion U (γ z ) = γ z . So U (Rfl γz ) = Rfl γ z . From U (z) = z, it follows that Rfl γz ∩ Rfl γ z = ∅. Let E γz be the set of closed edges of cl(K * ), which the billiard motion γ z reflects off of. In other words, E γz = {E an edge of cl(K * ) p ∈ E for some p ∈ Rfl γz }. Lemma 6.8 E γz = U (E γz ).
Proof. If p ∈ Rfl γz , then S m (p) ∈ U (Rfl γz ), for U (p) ∈ U (Rfl γz ), which implies R n0 ((U (p))) ∈ U (Rfl γz ), since U (Rfl γz ) is G-invariant. Hence S 0 (p) ∈ U (Rfl γz ). If p ∈ Rfl γz , then R −m (p) ∈ Rfl γz , since Rfl γz is G-invariant. So S 0 (R −m (p)) ∈ U (Rfl γz ), which implies R m S 0 (R −m (p)) ∈ U (Rfl γz ), because U (Rfl γz ) is G-invariant. So S m (Rfl γz ) ⊆ U (Rfl γz ). A similar argument shows that S m (U (Rfl γz )) ⊆ Rfl γz . Thus Rfl γz = S m (S m (Rfl γz )) ⊆ S m (U (Rfl γz )) ⊆ Rfl γz . perpendicular to the direction u k0 , which is first met by γ z and let P k0 be the meeting point. Let S k0 be the reflection in E k0 . The continuation of the motion γ z at P k0 is the horizontal line RS k0 (γ z ) in K * k0 . Recall that K * is the translation of K * by τ k0 . Since O k0 = τ k0 (0) is the center of K * k0 , the extended motion is the same as the motion U (γ z ) translated by τ k . Using a suitable sequence of Figure 14. (left) The billiard motion g z in the stellated regular 3-gon K ⇤ 1,1,1 meets the edge 0, is reflected in this edge by S 0 , and then is rotated by R. This gives an extended motion RS 0 g z , which is a straight line that is the same as reflecting g z by U and then translating by t 0 .
reflections in the edges of a suitable K * k0···k followed by a rotation R, which gives rise to a reflection U and a translation in T corresponding to their origins, we can extend s to a smooth straight line λ in C \ V + , see figure 14. The line λ is a geodesic in (C \ V + , γ| C\V + ), which in K * has image λ ν(z) under the G-orbit map that is a smooth geodesic on ( S reg , γ). The geodesic ν(λ) starts at ν(z). Thus the smooth geodesic λ ν(z) and the possibly broken geodesic ν(λ) are equal. In other words, ν(λ) is a smooth geodesic.
Thus the affine orbit space S reg = (C \ V + )/G with flat Riemannian metric γ is the affine analogue of the Poincaré model of the affine Riemann surface S reg as an orbit space of a discrete subgroup of PGl(2, C) acting on the unit disk in C with the Poincaré metric.