Open Math. 2020; 18:177–181

Open Mathematics

Research Article

Oğuzhan Demirel* The hyperbolic of type (ϵ, n) and Möbius transformations https://doi.org/10.1515/math-2020-0015 Received October 14, 2019; accepted February 5, 2020

Abstract: An n-sided hyperbolic of type (ϵ, n) is a hyperbolic polygon with ordered interior π ϵ θ θ ... θ π ϵ ϵ π θ π 2 + , 1, 2, , n−2, 2 − , where 0 < < 2 and 0 < i < satisfying

n X−2  π   π  θi + + ϵ + − ϵ < (n − 2)π 2 2 i=1 θ θ π i n θ π ϵ π θ π ϵ π and i + i+1 ≠ (1 ≤ ≤ − 3), 1 + ( 2 + ) ≠ , n−2 + ( 2 − ) ≠ . In this paper, we present a new characterization of Möbius transformations by using n-sided hyperbolic polygons of type (ϵ, n). Our proofs are based on a geometric approach.

Keywords: conformal mapping, Möbius transformations, hyperbolic polygons

MSC 2010: 30C20, 30C35, 30F45, 51M10, 51M15

1 Introduction

f → f z az+b a b c d ∈ ad bc A Möbius transformation : C C is a map dened by ( ) = cz+d , where , , , C with − ≠ 0. They are the automorphisms of extended complex plane C and dene the Möbius transformation group M(C) with respect to composition. Möbius transformations are also directly conformal homeomorphisms of C onto itself and they have beautiful properties. For example, a map is Möbius if and only if it preserves cross ratios. As for geometric aspect, circle-preserving is another important characterization of Möbius transformations. The following result is one of the most famous theorems for Möbius transformations:

Theorem 1. [1] If f : C → C is a circle preserving map, then f is a Möbius transformation if and only if f is a bijection.

f z az+b a b c d ∈ ad bc The transformations ( ) = cz+d , where , , , C with − ≠ 0 are known as conjugate Möbius transformations of C. It is easy to see that each conjugate Möbius transformation f is the composition of complex conjugation with a Möbius transformation. Since the complex conjugate transformation and Möbius transformations are homeomorphisms of C onto itself (complex conjugation is given by reection in the plane through R∪{∞}), conjugate Möbius transformations are homeomorphisms of C onto itself. Notice that the composition of a conjugate Möbius transformation with a Möbius transformation is a conjugate Möbius transformation and composition of two conjugate Möbius transformations is a Möbius transformation. There is topological distinction between Möbius transformations and conjugate Möbius transformations: Möbius transformations preserve the orientation of C, whereas conjugate Möbius transformations reverse it. To see

*Corresponding Author: Oğuzhan Demirel: Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey; E-mail: [email protected]

Open Access. © 2020 Oğuzhan Demirel, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 178 Ë Oğuzhan Demirel more details about conjugate Möbius transformations, we refer the reader to [2]. The following denitions are well known and fundamental in hyperbolic .

Denition 2. [3] A Lambert is a hyperbolic quadrilateral with ordered interior angles π , π , π and 2 2 2 θ, where θ π . 0 < < 2

Denition 3. [3] A is a hyperbolic quadrilateral with ordered interior angles π , π , θ, θ, 2 2 where θ π . 0 < < 2

A Möbius invariant property is naturally related to . To see the characteristics of Möbius transformations involving Lambert and Saccheri quadrilaterals, we refer the reader to [4]. Moreover, there are many characterizations of Möbius transformations by using various hyperbolic polygons; see, for instance, [5–7]. In [8, 9], O. Demirel presented some characterizations of Möbius transformations by using new classes of geometric hyperbolic objects called “degenerate Lambert quadrilaterals”and “degenerate Saccheri quadri- laterals”, respectively.

Denition 4. [9] A degenerate is a hyperbolic convex quadrilateral with ordered interior angles π ϵ, π , π ϵ, θ, where θ π and ϵ π θ . 2 + 2 2 − 0 < < 2 0 < < 2 − 2

Theorem 5. [9] Let f : B2 → B2 be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves all ϵ-Lambert quadrilaterals.

Denition 6. [8] A degenerate Saccheri quadrilateral is a hyperbolic convex quadrilateral with ordered angles π ϵ, π ϵ, θ θ, where θ π and ϵ π θ . 2 − 2 + , 0 < < 2 0 < < 2 − 2

Theorem 7. [8] Let f : B2 → B2 be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves all ϵ-Saccheri quadrilaterals.

In the theorems above, B2 is the open unit disc in the complex plane. Naturally, one may wonder whether the counterpart of Theorem 7 exists for hyperbolic polygons instead of using degenerate Saccheri quadrilaterals. Before giving the armative answer of this question let us state the following denition:

Denition 8. Let n be a positive integer satisfying n ≥ 5. An n-sided hyperbolic polygon of type (ϵ, n) is a convex hyperbolic polygon with ordered interior angles π ϵ θ θ ... θ π ϵ or π ϵ θ θ ... θ π ϵ, 2 + , 1, 2, , n−2, 2 − 2 − , 1, 2, , n−2, 2 + where ϵ π and θ π satisfying 0 < < 2 0 < i < n X−2  π   π  θi + + ϵ + − ϵ < (n − 2)π, 2 2 i=1 and θ θ π ( i n ), θ π ϵ π, θ π ϵ π. Notice that the sides of the n-sided hyperbolic i + i+1 ≠ 1 ≤ ≤ −3 1 +( 2 + ) ≠ n−2 +( 2 − ) ≠ polygon of type (ϵ, n) we mentioned here are hyperbolic line segments.

The existence of n-sided hyperbolic polygons of type (ϵ, n) is clear by the following result:

Lemma 9. [3] Let (θ1, θ2, ··· , θn) be any ordered n-tuple with 0 ≤ θj < π, j = 1, 2, ... , n. Then there exists a polygon P with interior angles θ θ ... θ , occurring in this order around ∂P, if and only if Pn θ n π. 1, 2, , n i=1 i < ( −2)

This paper presents a new characterization of Möbius transformations by use of mappings which preserve n-sided hyperbolic polygons of type (ϵ, n). To do so, we need Carathéodory’s theorem which plays a major role in our results. C. Carathéodory [10] proved that every arbitrary one-to-one correspondence between the points 0 of a circular disc C and a bounded point set C such that which circles lying completely in C are transformed 0 into circles lying in C must always be either a Möbius transformation or a conjugate Möbius transformation. The hyperbolic polygons of type (ϵ, n) and Möbius transformations Ë 179

0 Throughout the paper we denote by X the image of X under f, by [P, Q] the geodesic segment between points P and Q, by PQ the geodesic through points P and Q, by PQR the hyperbolic with vertices P, Q and R, by ∠PQR the between [P, Q] and [P, R] and by dH(P, Q) the hyperbolic distance between points 2 P Q B2 {z ∈ |z| } ds2 4|dz| and . We consider the hyperbolic plane = C : < 1 with length dierential = (1−|z|2)2 . The Poincaré disc model of hyperbolic geometry is built on B2; more precisely the points of this model are points of B2 and the hyperbolic lines of this model are Euclidean semicircular arcs that intersect the boundary of B2 orthogonally including diameters of B2. Given two distinct hyperbolic lines that intersect at a point, the measure of the angle between these hyperbolic lines is dened by the Euclidean tangents at the common point.

2 A characterization of Möbius transformations by use of hyperbolic polygons of type (ϵ, n)

The assertion f preserves n-sided hyperbolic polygons A1A2 ··· An of type (ϵ, n), n ≥ 5, with ordered interior π ϵ θ θ ... θ π ϵ A A A f n angles 2 − , 1, 2, , n−2, 2 + means that the image of 1 2 ··· n under is again an -sided hyperbolic A0 A0 A0 π ϵ θ0 θ0 ... θ0 π ϵ P polygon 1 2 ··· n with ordered interior angles 2 − , 1, 2, , n−2, 2 + and if is a point on any side 0 0 0 0 of A1A2 ··· An, then P is a point on any side of A1A2 ··· An.

Lemma 10. Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all ϵ π . Then f is injective. 0 < < 2

2 Proof. Let A1 and A2 be two distinct points in B . It is clear that there exists an (2n − 4)-sided hyperbolic (n > 4, n ∈ N), say A1A2 ··· A2n−4. By β denote the interior angles of A1A2 ··· A2n−4. Let M and N be the midpoints of [A2n−4, A1] and [An−2, An−1], respectively. Then the hyperbolic polygons MA1A2 ··· An−2N and NAn−1An ··· A2n−4M are n-sided hyperbolic polygons satisfying MN⊥A1A2n−4 and MN⊥An−2An−1. Let P be a point on [M, A1] and let Q be a point on [N, An−1] satisfying dH(P, A1) = dH(Q, An−1). By ψ denote the angle ∠QPA1. Since A1A2A3 ··· A2n−4 is an (2n − 4)-sided hyperbolic regular PQA ψ ψ π ψ π α ψ π ψ π α polygon, we immediately get ∠ n−1 = . If > 2 let’s denote = 2 + and if < 2 let’s denote = 2 − . Hence we see that PA1A2 ··· An−2Q is an n-sided hyperbolic polygon of type (α, n). By assumption, we obtain 0 0 0 0 0 0 0 that P A1A2 ··· An−2Q is also an n-sided hyperbolic polygon of type (α, n), which implies A1 ≠ A2. Thus f is injective.

Lemma 11. Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all ϵ π . Then f preserves the collinearity and betweenness property of the points. 0 < < 2

Proof. Let P and Q be two distinct points in B2 and assume that S is an interior point of [P, Q]. Let ∆ be the set of all n-sided hyperbolic polygons of type (ϵ, n) such that the points P and Q are two adjacent vertices of these hyperbolic polygons. Then S belongs to all elements of ∆. By the property of f, the images of the elements of ∆ 0 0 are n-sided hyperbolic polygons of type (ϵ, n) whose vertices contain P and Q . Moreover, the images of the 0 0 0 0 0 elements of ∆ must contain S . Since f is injective by Lemma 10, we get P ≠ S ≠ Q . Therefore, S must be an 0 0 interior point of [P , Q ], which implies that f preserves the collinearity and betweenness of the points.

Lemma 12. Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all ϵ π . Then f preserves the angles π ϵ and π ϵ. 0 < < 2 2 + 2 −

Proof. A A A n ϵ n A A A π Let 1 2 ··· n be an -sided hyperbolic polygon of type ( , ) such that ∠ n 1 2 = 2 − ϵ A A A θ i n A A A π ϵ A , ∠ i−1 i i+1 = i (2 ≤ ≤ − 1) and ∠ n−1 n 1 = 2 + . Let us denote the midpoint of 1 and 2 An by M. Clearly, the hyperbolic line A1A2 and the complex unit disc B meet at two points, say P and Q. Assume A1 ∈ [P, A2] and ∠MA2Q = ρ. Let X be a point on [P, A2] moving from P to A2. It is easy to 180 Ë Oğuzhan Demirel

see that if X moves from P to A2, the angle ∠MXQ increases continuously from 0 to ρ. Let H be a point A A MHA π A A S on [ 1, 2] such that ∠ 2 < 2 . Now take a point on the hyperbolic line n n−1, say , satisfying An ∈ [S, An−1] and dH(S, An) = dH(H, A1). It is easy to see that dH(M, A1) = dH(M, An), ∠SAn M = ∠HA1M, and dH(A1, H) = dH(An , S) hold, which implies that the hyperbolic HA1M and SAn M are congruent triangles by the hyperbolic side-angle-side theorem. Hence we get ∠A1MH = ∠An MS and this yields that H M S MHA MSA MHA π the points , and must be collinear and ∠ 1 = ∠ n. Since ∠ 2 < 2 , we may assume the MHA π α α π MSA π α α representation ∠ 2 = 2 − , where 0 < < 2 , which implies ∠ n = 2 + . Notice that must be less ϵ π ϵ π α HA A S n than since 2 − < 2 − . Therefore, one can easily see that 2 ··· n−1 is an -sided hyperbolic polygon of type (α, n). By the property of f, the images of the hyperbolic polygons A1A2 ··· An and HA2 ··· An−1S are 0 0 0 n-sided hyperbolic polygons of type (ϵ, n) and type (α, n), respectively. The hyperbolic polygons A1A2 ··· An H0A0 A0 S0 n A0 A0 A0 π ϵ S0H0A0 π α and 2 ··· n−1 have − 2 common angles, which implies ∠ n 1 2 = 2 ± and ∠ 2 = 2 ± . A0 A0 A0 π ϵ Lemma 11 H0 ∈ A0 A0 M0 ∈ A0 A0 M0 ∈ H0 S0 Now assume ∠ n 1 2 = 2 + . By , we get [ 1, 2], [ 1, n] and [ , ]. Since S0H0A0 π α α ϵ ∠ 2 = 2 ± and < hold, we get that the sum of the measures of interior angles of the hyperbolic M0A0 H0 A0 M0H0 π ϵ π α π triangle 1 is ∠ 1 + ( 2 + ) + ( 2 ± ), which is greater then . This is not possible in hyperbolic A0 A0 A0 π ϵ A0 A0 A0 π ϵ geometry and so we get ∠ n 1 2 = 2 − , which yields ∠ n−1 n 1 = 2 + .

Lemma 13. Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all ϵ π . Then f preserves the hyperbolic distance. 0 < < 2

Proof. Let P and Q be two distinct points in B2. Take a point S such that PQS forms a hyperbolic equilateral β PQS QSP SPQ β β π β π α triangle. By denote its angles ∠ = ∠ = ∠ := . Since < 2 let’s use the representation = 2 − α π By Lemma 9 n α n A A A with 0 < < 2 . , there exists an -sided hyperbolic polygon of type ( , ), say 1 2 ··· n, such A A A γ A A A γ ... A A A γ A A A π α A A A π α that ∠ 1 2 3 = 2, ∠ 2 3 4 = 3, , ∠ n−2 n−1 n = n−1, ∠ n−1 n 1 = 2 + and ∠ n 1 2 = 2 − . Then the angle ∠An A1A2 of the hyperbolic polygon A1A2 ··· An can be moved to the point P by using an appropriate hyperbolic isometry g such that g(A2) ∈ [P, S] (or S ∈ [P, g(A2)]) and g(An) ∈ [P, Q] (or Q ∈ P g A f π ϵ π ϵ n ϵ n [ , ( n)]). Since preserves the angles 2 + and 2 − of -sided hyperbolic polygons of type ( , ) for all ϵ π Lemma 12 0 < < 2 by , we get

π 0 0 0 0 0 0 − α = ∠SPQ = ∠g(An)g(A1)g(A2) = ∠g(An) g(A1) g(A2) = ∠S P Q , 2 0 0 0 0 0 0 which implies ∠PQS = ∠P Q S and ∠QSP = ∠Q S P . Because of the fact that the angles at the vertices of a 0 0 determine its lengths, we get dH(P, Q) = dH(P , Q ); see [11, 12].

Corollary 14. Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all ϵ π . Then f preserves the measures of all angles. 0 < < 2

Theorem 15. Let f : B2 → B2 be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type ϵ n for all ϵ π . ( , ) 0 < < 2

Proof. The “only if” part is clear since f is an isometry. Conversely, we may assume that f preserves all n-sided hyperbolic polygons of type (ϵ, n) in B2 and f (O) = O by composing an hyperbolic isometry if necessary. Let us 2 0 take two distinct points in B and denote them by x, y. By Lemma 13, we immediately get dH(O, x) = dH(O, x ) 0 0 0 and dH(O, y) = dH(O, y ), namely |x| = |x | and |y| = |y |, where | · | denotes the Euclidean norm. Therefore, 0 0 we get |x − y| = |x − y | since f preserves angular sizes by Corollary 14. As

0 0 0 0 0 0 2hx, yi = |x|2 + |y|2 − |x − y|2 = |x |2 + |y |2 − |x − y |2 = 2hx , y i, f preserves the Euclidean inner-product this implies that f is a restriction of an orthogonal transformation on B2, that is, f is a Möbius transformation or a conjugate Möbius transformation by Carathéodory’s theorem.

Corollary 16. Let f : B2 → B2 be a conformal (angle preserving with sign) surjective transformation. Then f is a Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type ϵ n for all ϵ π . ( , ) 0 < < 2 The hyperbolic polygons of type (ϵ, n) and Möbius transformations Ë 181

Corollary 17. Let f : B2 → B2 be an angle reversing surjective transformation. Then f is a conjugate Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type ϵ n for all ϵ π . ( , ) 0 < < 2

Acknowledgement: The author would like to thank the anonymous reviewers for their careful, constructive and insightful comments in relation to this work.

References

[1] Janos Aczel and Michael A. Mckiernan, On the characterization of plane projective and complex Möbius transformations, Math. Nachr. 33 (1967), 315–337, DOI 10.1002/mana.19670330506. [2] Gareth A. Jones and David Singerman, Complex functions: An algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987. [3] Alan F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983. [4] Yang Shihai and Fang Ainong, A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006), no. 2, 660–664, DOI 10.1016/j.jmaa.2005.05.082. [5] Liu Jing, A new characteristic of Möbius transformations by use of polygons having type A, J. Math. Anal. Appl. 324 (2006), no. 1, 281–284, DOI 10.1016/j.jmaa.2005.12.015. [6] Oğuzhan Demirel and Emine Soytürk Seyrantepe, A characterization of Möbius transformations by use of hyperbolic regular polygons, J. Math. Anal. Appl. 374 (2011), no. 2, 566–572, DOI 10.1016/j.jmaa.2010.08.049. [7] Oğuzhan Demirel, A characterization of Möbius transformations by use of hyperbolic regular star polygons, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), no. 1, 13–19, https://acad.ro/sectii2002/proceedings/doc2013-1/03- Demirel.pdf. [8] Oğuzhan Demirel, Degenerate Lambert quadrilaterals and Möbius transformations, Bull. Math. Soc. Sci. Math. Roumanie 61 (109) (2018), no. 4, 409–415, https://ssmr.ro/bulletin/volumes/61-4/node6.html. [9] Oğuzhan Demirel, Degenerate saccheri quadrilaterals, Möbius transformations and conjugate Möbius transformations, Int. Elect. Journ. of Geometry 10 (2017), no. 2, 32–36, https://dergipark.org.tr/tr/download/article-le/680419. [10] Constantin Carathéodory, The most general transformations of plane regions which transform circles into circles, Bull. Am. Math. Soc. 43 (1937), 573–579, DOI 10.1090/S0002-9904-1937-06608-0. [11] Abraham Albert Ungar, Analytic hyperbolic geometry. Mathematical foundations and applications, World Scientic Publish- ing Co. Pte. Ltd., Hackensack, NJ, 2005. [12] Abraham Albert Ungar, The hyperbolic and Möbius transformations, Banach J. Math. Anal. 1 (2007), no. 1, 101–116, DOI 10.15352/bjma/1240321560.