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A Characteristic of Gyroisometries in Möbius Gyrovector Spaces

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A Characteristic of Gyroisometries in Möbius Gyrovector Spaces WN North-Western European Journal of Mathematics M EJ A characteristic of gyroisometries in Möbius gyrovector spaces Oğuzhan Demırel1 Received: October 18, 2019/Accepted: January 30, 2019/Online: March 1, 2020 Abstract Hugo Steinhaus (1966a, b) has asked whether inside each acute angled triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. In this paper, we prove that f : D D is a gyroisometry (hyperbolic isometry) if, and only if it is a continuous mapping! that preserves the partition of a gyrotriangle (hyperbolic triangle) asked by Hugo Steinhaus. Keywords: Gyrogroups, Möbius Gyrovector Spaces, Gyroisometry. msc: 51B10, 30F45, 20N05, 51M09, 51M10, 51M25. 1 Introduction A Möbius transformation f : C C is a mapping of the form w = [ f1g ! [ f1g (az + b)=(cz + d) satisfying ad bc , 0, where a,b,c,d C. The set of all Möbius transformations Möb(C )−is a group with respect2 to the composition and any f Möb(C ) is conformal,[ f1g i.e. it preserves angles. Let us define 2 [ f1g Ω = S C : S is an Euclidean circle or a Euclidean line . f ⊂ [ f1g [ 1g It is well known that if C Ω and f Möb(C ), then f (C) Ω. There are well- known elementary proofs2 that if f is2 a continuous[ f1g injective map2 of the extended complex plane C that maps circles into circles, then f is Möbius. In addition to this a map is Möbius[ f1g if, and only if it preserves cross ratios. In function theory, it is known that a function f is Möbius if, and only if the Schwarzian derivative of f vanishes when f 0(z) , 0. Using this differential criterion, H. Haruki and T.M. Rassias2 proved that if f is meromorphic and preserves Apollonius quadrilaterals, then f is Möbius. 1Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey 2Haruki and Rassias, 1998, “A new characteristic of Möbius transformations by use of Apollonius quadrilaterals”. 107 A characteristic of gyroisometries in Möbius gyrovector spaces O. Demırel The Möbius invariant property is also naturally related to hyperbolic geometry. For instance, in the hyperbolic plane, Möbius transformations can be characterized by Lambert (and Saccheri) quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals (or Saccheri quadrilaterals to Saccheri quadrilaterals) must be Möbius, see Yang and Fang (2006a), Yang and Fang (2006b). Moreover, in literature there are many characterizations of Möbius transformations by using of triangular domains3, regular hyperbolic polygons4, hyperbolic regular star polygons5, polygons having type A6, and others. 2 Möbius transformations of the Disc, Möbius Addi- tion and Möbius Gyrovector Spaces Let us denote the complex open unit disc (centered at origin) in C by D and z0 be an element of C. Clearly the mapping z + z f (z) = eiθ 0 , θ R 1 + z0z 2 is a Möbius transformation satisfying f (D) = D. L. Ahlfors7 proved that the most general Möbius transformation of D is given by the polar decomposition iθ z0 + z iθ z e = e (z0 z). ! 1 + z0z ⊕ It induces the Möbius addition “ ” in the disc, allowing the Möbius transformation of the disc to be viewed as Möbius⊕ left gyrotranslation z0 + z z z0 z = ! ⊕ 1 + z0z followed by rotation. Here θ R, z0 D and Möbius substraction “ ” is defined by a z = a ( z). Clearly z z =2 0 and 2z = z.The groupoid (D, ) is not a group since the groupoid⊕ − operation “ ” is not associative. − In addition to⊕ this the commutative property does not hold. However,⊕ the groupoid (D, ) has a group-like structure. The breakdown of commutativity in Möbius addition⊕ is “repaired” by the intro- duction of gyration, gyr : D D Aut(D, ) × ! ⊕ 3Li and Wang, 2009, “A new characterization for isometries by triangles”. 4Demirel and Seyrantepe, 2011. 5Demirel, 2013, “A characterization of Möbius transformations by use of hyperbolic regular star polygons”. 6Liu, 2006, “A new characteristic of Möbius transformations by use of polygons having type A”. 7Ahlfors, 1978, Complex analysis: An introduction to the theory of analytic functions of one complex variable. 108 2. Möbius transformations of the Disc, Möbius Addition and Möbius Gyrovector Spaces given by the equation a b 1 + ab gyr [a,b] = ⊕ = (1) b a 1 + ab ⊕ where Aut(D, ) is the automorphism group of the groupoid (D, ). Therefore, the gyrocommutative⊕ law of Möbius addition follows from the definition⊕ of gyration in (1), ⊕ a b = gyr [a,b](b a). (2) ⊕ ⊕ Coincidentally, the gyration gyr[a,b] that repairs the breakdown of the commutative law of in (2), repairs the breakdown of the associative law of as well, giving rise to the respective⊕ left and right gyroassociative laws ⊕ a (b c) = (a b) gyr [a,b]c (a⊕ b)⊕ c = a ⊕(b ⊕gyr [b,a]c) ⊕ ⊕ ⊕ ⊕ for all a,b,c D. 2 Definition 1 – A groupoid (G, ) is a gyrogroup if its binary operation satisfies the following axioms ⊕ (G1) For each a G, there is an element 0 G such that 0 a = a. (G2) For each a 2 G, there is an element b 2 G such that b ⊕a = 0. (G3) For all a,b 2 G, there is an automorphism2 gyr[a,b] ⊕Aut(G, ) such that 2 2 ⊕ a (b c) = (a b) gyr[a,b]c ⊕ ⊕ ⊕ ⊕ (G4) For all a,b G, gyr [a,b] = gyr [a b,b]. 2 ⊕ Additionally, if the binary operation “ ” obeys the gyrocommutative law G5, then (G, ) is called a gyrocommutative gyrogroup.⊕ ⊕ (G5) For all a,b G, a b = gyr [a,b](b a). 2 ⊕ ⊕ Clearly, with these properties, one can now readily check that the Möbius complex disc groupoid (D, ) is a gyrocommutative gyrogroup. We refer readers to Ungar (2001, 2008) for more⊕ details about gyrogroups. Identifying complex numbers of the complex plane C with vectors of the Eu- clidean plane R2 in the usual way: C u = u + iu = (u ,u ) = u R2. 3 1 2 1 2 2 Then the equations u v = Re(uv) (3) u· = u . k k j j 109 A characteristic of gyroisometries in Möbius gyrovector spaces O. Demırel give the inner product and the norm in R2, so that Möbius addition in the disc D of n o C becomes Möbius addition in the disc R2 = v R2 : v < 1 of R2. In fact we get 1 2 k k from (3) u + v u v = _ ⊕ 1 + uv _ (1 + uv)(u + v) = _ _ (1 + uv)(1 + uv) _ _ 1 + uv+uv + v 2 u + 1 u 2 v = _ j j_ − j j (4) 1 + uv+uv + u 2 v 2 j j j j 1 + 2u v+ v 2 u+ 1 u 2 v = · k k − k k 1 + 2u v+ u 2 v 2 · k k k k = u v ⊕ D R2 for all u,v and all u,v 1. Let V =2 (V ,+, ) be any inner-product2 space and · V = v V : v < s s f 2 k k g V V be the open ball of with radius s > 0. Möbius addition in s is motivated by (4) and it is given by the equation 1 + 2=s2 u v + 1=s2 v 2 u + 1 1=s2 u 2 v u v = · k k − k k (5) ⊕ 1 + (2=s2)u v + 1=s4 u 2 v 2 · k k k k V where and are the inner product and norm that the ball s inherits from its space V· . Withoutk·k loss of generality, we may assume that s = 1 in (5). However we prefer to keep s as a free positive parameter in order to exhibit the results that in V the limit as s , the ball s expands to the whole of its real inner product space V , and Möbius! 1addition reduces to vector addition + in V , i.e., ⊕ lim u v = u + v s −!1 ⊕ and V V lim s = . s !1 Möbius scalar multiplication “ ” is given by the equation ⊗ 1 v r v = s tanh r tanh− v =s (6) ⊗ k k v k k 110 2. Möbius transformations of the Disc, Möbius Addition and Möbius Gyrovector Spaces R V where r , u,v s , v , 0 and r 0 = 0. Möbius scalar multiplication possesses the following2 properties:2 ⊗ (P1) n v = v v v,(n-term) ⊗ ⊕ ⊕ ··· ⊕ (P2) (r + r ) v = r v r v scalar distribute law 1 2 ⊗ 1 ⊗ ⊕ 2 ⊗ (P3) (r r ) v = r (r v) scalar associative law 1 2 ⊗ 1 ⊗ 2 ⊗ (P4) r (r v r v) = r (r v) r (r v) monodistribute law ⊗ 1 ⊗ ⊕ 2 ⊗ ⊗ 1 ⊗ ⊕ ⊗ 2 ⊗ (P5) r v = r v homogenity property k ⊗ k j j ⊗ k k r v v (P6) jrj⊗v = v scaling property k ⊗ k k k (P7) gyr[a,b](r v) = r gyr[a,b]v gyroautomorphism property ⊗ ⊗ (P8)1 v = v multiplicative unit property ⊗ V Definition 2 (Möbius gyrovector spaces) – Let ( s, ) be a Möbius gyrogroup equipped with scalar multiplication . The triple (V , , ) is called⊕ a Möbius gyrovector space. ⊗ s ⊕ ⊗ Definition 3 – The Möbius gyrodistance between the points A,B in Möbius gy- rovector space (V , ) is given by the equation s ⊕ d(A,B) = A B . k k The Möbius gyrodistance function, in general gyrodistance function, gives rise to a gyrotriangle inequality which involves a gyroaddition . In contrast, the familiar hyperbolic distance function in the literature is designed⊕ so as to give rise to a triangle inequality which involves the addition +. The connection between the gyrodistance function and the standard hyperbolic distance function is described in Ungar (1999).
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