The Hyperbolic Derivative in the Poincare Ball Model of Hyperbolic Geometry

Transcription

1 Ž. Journal of Mathematical Analysis and Applications 254, doi:0.006jmaa , available online at on The Hyperbolic Derivative in the Poincare Ball Model of Hyperbolic Geometry Graciela S. Birman Departamento de Matematica, Facultad de Ciencias Exactas, Uniersidad Nacional del Centro de la Proincia de Buenos Aires, Pinto 399, 7000 Tandil, Argentina and Abraham A. Ungar Department of Mathematics, North Dakota State Uniersity, Fargo, North Dakota abraham Submitted by T. M. Rassias Received May 5, 2000 The generic Mobius transformation of the complex open unit disc induces a binary operation in the disc, called the Mobius addition. Following its introduction, the extension of the Mobius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincare ball model of hyperbolic geometry. The Mobius gyrovector spaces that emerge provide the setting for the Poincare ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Mobius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry. 200 Academic Press Partially supported by Consejo Nacional de Investigaciones Cientificas y Tecnicas of Republica Argentina X0 $35.00 Copyright 200 by Academic Press All rights of reproduction in any form reserved.

2 322 BIRMAN AND UNGAR. INTRODUCTION In the 20th century the notions of group and vector space dominated analysis, geometry, and physics to the present days. However, their generalization to gyrogroups and gyrovector spaces, which sprung from the soil of Einstein s special theory of relativity, does not seem to have fired the interest of physicists to the same extent despite their compelling application in hyperbolic geometry and in relativistic physics. Thus, despite the fascination of gyrocommutative gyrogroups Žwhich, following 6, are also known as K-loops. in non-associative algebra 4 for over a decade, it is fair to say that they still await universal acceptance. This is not to say that there have not been valiant attempts to find appropriate uses for them. One can point to their impact on special relativity theory and hyperbolic geometry 97. The theories of gyrogroups and gyrovector spaces provide a new avenue for investigation, leading to a new approach to hyperbolic geometry 2 and, subsequently, to new Ž as yet to be discovered. physics. The discovery of the first gyrogroup 7 followed the exposition of the mathematical regularity that the Thomas precession stores 5. The Thomas precession of relativity physics is a rotation that has no classical counterpart. It has been extended in 9 by abstraction to the so-called Thomas gyration which, in turn, suggests the prefix gyro that we extensively use to emphasize analogies with classical notions. Thomas gyration is an isometry of hyperbolic geometry that any two points of the geometry generate, enabling novel analogies shared by Euclidean and hyperbolic geometry to be exposed. The novel analogies, in turn, allow the unification of Euclidean and hyperbolic geometry and trigonometry 4, 5. The aim of this article is Ž. i to extend the differential operation in vector spaces to a differential operation in gyrovector spaces; and Ž ii. to study its application to geodesics. 2. GYROGROUPS AND GYROVECTOR SPACES Gyrogroups are generalized groups that share remarkable analogies with groups 9. In full analogy with groups Ž. gyrogroups are classified into gyrocommutative gyrogroups and non-gyrocommutative gyrogroups; and Ž. 2 some gyrocommutative gyrogroups admit scalar multiplication, turning them into gyrovector spaces; Ž. 3 gyrovector spaces, in turn, provide the setting for hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry, thus enabling the two geometries to be unified.

3 THE POINCARE BALL MODEL 323 In the study of Mobius transformations patterns and interesting concepts emerge from time to time as, for instance, in 2, 3, 6. A most intriguing motivation for the introduction of the notion of a gyrogroup is, indeed, provided by the Mobius transformation group of the complex open unit disc. The most general Mobius transformation of the complex open unit disc z : z 4 in the complex z-plane, 3, a z i i z e e Ž a z. Ž 2.. az induces the Mobius addition in the disc, allowing the Mobius transformation of the disc to be viewed as a Mobius left gyrotranslation a z z a z Ž 2.2. az followed by a rotation. Here is a real number, a, z, and a is the complex conjugate of a. Mobius addition is neither commutative nor associative. The breakdown of commutativity in Mobius addition is repaired by the introduction of gyration, gyr : AutŽ,., given by the equation a b ab gyra, b, Ž 2.3. b a ab where AutŽ,. is the automorphism group of the groupoid Ž,..We recall that a groupoid Ž G,. is a nonempty set G with a binary operation, and an automorphism of a groupoid Ž G,. is a bijective self-map of the groupoid G which respects its binary operation. The set of all automorphisms of a groupoid Ž G,. forms a group under bijection composition, denoted AutŽ G,.. The gyrocommutatie law of Mobius addition follows from the definition of gyr in Ž 2.3., a b gyra, bž b a.. Ž 2.4. Coincidentally, the gyration gyra, b that repairs the breakdown of the commutative law of in Ž 2.4., repairs the breakdown of the associative law of as well, giving rise to the respective left and right gyroassociatie

4 324 BIRMAN AND UNGAR laws a Ž b z. Ž a b. gyra, bz Ž a b. z a Ž b gyrb, az. Ž 2.5. for all a, b, z. Guided by analogies with groups we take key features of the Mobius addition as a model of a gyrogroup, obtaining the following DEFINITION 2. Ž Gyrogroups.. The groupoid Ž G,. is a gyrogroup if its binary operation satisfies the following axioms. In G there is at least one element, 0, called a left identity, satisfying Ž G. 0 a a, Left Identity for all a G. There is an element 0 G satisfying axiom Ž G. such that for each a in G there is an element a in G, called a left inverse of a, satisfying Ž G2. a a 0, Left Inverse. Moreover, for any a, b, z G there exists a unique element gyra, bz G such that Ž G3. a Ž b z. Ž a b. gyra, bz, Left Gyroassociative Law. If gyra, b denotes the map gyra, b : G G given by z gyra, bz then Ž G4. gyra, b AutŽ G,., Gyroautomorphism and gyra, b is called the Thomas gyration, or the gyroautomorphism of G, generated by a, b G. Finally, the gyroautomorphism gyra, b generated by any a, b G satisfies Ž G5. gyra, b gyra b, b, Left Loop Property. DEFINITION 2.2 Ž Gyrocommutative Gyrogroups.. A gyrogroup Ž G,. is gyrocommutative if it satisfies Ž G6. a b gyra, bž b a., Gyrocommutative Law. Grouplike gyrogroup theorems that follow from Definitions 2. and 2.2 are presented in 9. These theorems ensure, for instance, that there exists a unique identity Ž which is both left and right. and a unique inverse Žwhich is both left and right.. Gyrocommutativity, for instance, is equivalent to the validity of the automorphic inverse law, as shown in the following Ž. THEOREM 2.3. A gyrogroup G, is gyrocommutatie if and only if Ž G7. Ž a b. a b, Automorphic Inerse Law.

5 THE POINCARE BALL MODEL 325 Furthermore, the left gyroassociative law and the left loop property are accompanied by right counterparts, Ž G8. Ž a b. z a Žb gyrb, az,. Right Gyroassociative Law Ž G9. gyra, b gyra, b a, Right Loop Property and the left cancellation law is valid, Ž G0. a Ž a b. b, Left Cancellation Law. 3. THE MOBIUS GYROGROUPS AND GYROVECTOR SPACES DEFINITION 3. Ž The Mobius Addition.. Let be a real inner product space, and let v : v c4 c be the open c-ball of of radius c 0. Mobius addition M in the ball c is a binary operation in c given by the equation 8 u v Ž Ž. Ž.. Ž Ž.. 2 Ž 2c. u v Ž c 4. u v 2 2c u v c 2 v 2 u c 2 u 2 v M 2 2, Ž 3.. where and are the inner product and norm that the ball c inherits from its space. To justify calling M in Definition 3. a Mobius addition we will show in Ž. Ž below that M of 3. in two dimensions, when and 2, is equivalent to of Ž 2.2. c c M in the complex unit disc. For this sake we identify vectors in 2 with complex numbers in the usual way, u Ž u, u2. u iu2 u. Ž 3.2. The inner product and the norm in 2 then become the real numbers u u u v ReŽ u. 2 Ž 3.3. u u, where u is the complex conjugate of u. Ž. 2 2 Under the translation 3.3 of elements of the open unit disc c of into elements of the complex unit disc, the Mobius addition Ž 3.. in the

6 326 open ball 2 c c BIRMAN AND UNGAR of 2, with c for simplicity, takes the form u v Ž. Ž. 2u v v 2 u u 2 v 2u v u v M Ž. Ž. u u u u 2 2 u u u Ž u.ž u. u u u u Ž.Ž. u Ž 3.4. for all u, v 2 c and all u,, as desired. We will now use the notation M. Since Ž 3.. forms a most natural extension of the Mobius addition in Ž 2.2., the following theorem is expected 9. THEOREM 3.2 Ž Mobius Gyrogroups.. Let c be the c-ball of a real inner product space, and let be the Mobius addition Ž 3.. in c. Then the groupoid Ž,. is a gyrocommutatie gyrogroup Ž c called a Mobius gyrogroup.. DEFINITION 3.3 Ž Mobius Scalar Multiplication.. Let Ž,. c be a Mobius gyrogroup. The Mobius scalar multiplication r v v r is given by the equation r Ž v c. Ž v c. v r v c r r Ž v c. Ž v c. v v v c tanh r tanh, Ž 3.5 ž /. c v where r, v c, v 0; and r 0 0. The Mobius addition M and its associated scalar multiplication possess the following properties. For all r, r, r2 and a, b, u, v c, Ž V. v v Ž V2. Ž r r. 2 v r v r2 v, Scalar Distributive Law Ž V3. Ž rr. v r Ž r v., Scalar Associative Law 2 2 r

7 THE POINCARE BALL MODEL 327 Ž V4. r Ž r v r v. r Ž r v. r Ž r v. 2 2, Monodistributie Law Ž V5. Žr v. r v vv, Scaling Property Ž V6. r v r v, Homogeneity Property Ž V7. u v u v, Gyrotriangle inequality Ž V8. gyra, bž r v. r gyra, bv, Gyroautomorphism Property. We may note that, ambiguously, in Ž V7. is a binary operation in c and in c. The introduction of scalar multiplication satisfying properties Ž V. Ž V7. suggests the following DEFINITION 3.4 Ž Mobius Gyrovector Spaces.. Let Ž,. c be a Mobius gyrogroup equipped with scalar multiplication. The triple Ž,,. c is called a Mobius gyrovector space. 4. GEODESICS IN MOBIUS GYROVECTOR SPACES The unique Euclidean geodesic passing through two given points a and b of a vector space n can be represented by the expression a Ž a b. t, Ž 4.. t. It passes through a when t 0, and through b when t. In full analogy, the unique hyperbolic geodesic passing through two given points a and b of a Mobius gyrovector space Ž,,. c can be represented by the expression a Ž a b. t, Ž 4.2. t, where a a. It passes through a when t 0, and through b when t. The latter follows from the left cancellation law in any Ž. 2 gyrogroup 9. The graphs of 4.2 in two dimensions, c c, and in three dimensions, c 3 c, are shown in Figs. and 2. These are circles that intersect the boundary of the ball c orthogonally, and are recognized as the geodesics of the Poincare ball model of hyperbolic geometry in two and in three dimensions, respectively. The well known Poincare distance function d in the ball can be written in terms of Mobius addition as as explained in 0,. dž a, b. a b Ž 4.3.

8 328 BIRMAN AND UNGAR FIG.. The unique 2-dimensional Mobius geodesic Ž 4.2. that passes through the two given points a and b. FIG. 2. The unique 3-dimensional Mobius geodesic Ž 4.2. that passes through the two given points a and b.

9 THE POINCARE BALL MODEL 329 To emphasize analogies with Euclidean geometry, the geodesics Ž 4.2. are also called gyrolines. The gyrolines Ž 4.2. are geodesics relative to the Poincare metric Ž The Poincare metric induces a topology relative to which continuity and limits are defined enabling us to define the gyroderivative in a Mobius gyrovector space. 5. GYRODERIVATIVES: THE HYPERBOLIC, GYROVECTOR SPACES DERIVATIVES Guided by analogies with vector spaces, we define the gyroderivative. The effects of the gyroderivative are then explored by studying its application to parametrized geodesics. DEFINITION 5. Ž The Gyroderivative.. Let Ž,,. c be a Mobius gyrovector space, and let v : c be a map from the real line into the ball c. If the limit vž t. lim vž t. vž t h. 4 Ž 5.. h0 h exists for any t, we say that the map v is differentiable on, and that Ž. Ž. Ž. the gyroderivative or, hyperbolic derivative of v t is v t. The gyroderivative in a Mobius ball gyrovector space Ž,. c is closely related to the ordinary derivative in the vector space of the ball c. Indeed, in the limit of small neighborhood of any point of c, hyperbolic geometry reduces to Euclidean geometry. Accordingly, Ž. i the Mobius addition in reduces to the vector addition in, and Ž ii. c the Mobius scalar multiplication reduces to the scalar multiplication in near any point of. Hence, the gyroderivative vž. c t of the map v : c given by vž t. a b t Ž 5.2. t, a, b c is expected to be a Euclidean vector parallel to the Euclidean tangent line at any point vž t., t, of the geodesic vž t. 0 0.We will show in Ž 5.3. and, graphically, in Fig. 3 that this is indeed the case. Despite its close relation to the ordinary derivative in a vector space, the gyroderivative introduces simplicity and elegance when applied to geodesics.

10 330 BIRMAN AND UNGAR n n FIG. 3. Euclidean tangent lines at points of Mobius geodesics. vž t. a b t, t, n, 2, 3, are three points on the geodesic vž. n t a b t, parametrized by t. The Euclidean tangent line at any point vž. t of the geodesic is Euclidean parallel to the vector gyrvž. t, ab which, by Ž 5.3., equals vž t.. Shown are the tangent lines at the three points of the geodesic, vž t., vž t., vž t. 2 3, and their corresponding Euclidean parallel vectors in Ž 2 the Mobius disc,,., which is the Poincare disc model of hyperbolic geometry. c Calculating the gyroderivative of vž. t in Ž 5.2. and employing gyrogroup manipulations we have the following chain of equations, some of which are numbered for later reference, vž t. lim vž t. vž t h. h0 h 4 lim Ž a b t. a b Ž t h. h0 h Ž. 4 lim Ž a b t. a Ž b t b h. h0 h Ž 2. lim Ž a b t. h0 h 4 Ž a b t. gyra, b tž b h. 4

11 Ž 3. THE POINCARE BALL MODEL 33 lim gyra, b tž b h. h0 h Ž 4. 4 lim Ž gyra, b tb. h h0 h Ž 5. lim gyra, b tb h0 gyra, b tb 4 Ž 6. gyr a, a Ž a b t. b gyr a, a vž t. b Ž 7. gyr vž t., a vž t. b Ž 8. gyr vž t., a b. Ž 5.3. Ž. The derivation of 5.3 follows. Ž. follows from the scalar distributive law Ž V2.. Ž. 2 follows from the left gyroassociative law Ž G3.. Ž. 3 follows from the left cancellation law Ž G0.. Ž. 4 follows from the gyroautomorphism property Ž V8.. Ž. 5 follows from the scalar associative law Ž V3.. Ž. 6 is verified by applying the left cancellation law Ž G0.. Ž. 7 follows from a left loop and a left cancellation. Ž. 8 follows from a right loop. Ž. In 5.3 we have thus verified the gyrovector space identity vž t. Ž a b t. gyra b t, ab gyra, b tb Ž 5.4. according to which the gyroderivative of the parametric gyroline vž t. a b t Ž 5.5. is the gyrovector b gyrated by a gyroautomorphism. The parametric gyroline vž. t is shown in Fig. 3 where three of its points, vž t., vž t., and vž t. 2 3 corresponding to the parameter values t t, t 2, and t3 in, are emphasized. The Euclidean tangent lines of the gyroline at the three selected points vž t., vž t., and vž t. are shown as well as their 2 3

12 332 BIRMAN AND UNGAR corresponding Euclidean vectors gyrvž t., ab, gyrvž t., ab 2 and gyrvž t., a 3 b, to which they are respectively Euclidean parallel. Hence, the gyroderivative of a gyroline introduces elegance when noticing that the gyroderivative of the parametric gyroline is analogous to the derivative of a parametric line, where the analogy reveals itself in terms of a Thomas gyration. We now wish to explore another analogy, that gyrolines share with lines, to which the gyroderivative gives rise. Guided by analogies with Euclidean geometry, we define the tangent gyroline of the hyperbolic curve vž. t in Ž 5.2. at any point vž t., t being a constant, to be the gyroline 0 0 yž t. vž t0. vž t0. t, Ž 5.6. t. In Euclidean geometry the tangent line of a line at any of its points is identical with the line itself. We will show that this is the case for gyrolines as well. Substituting vž t. from Ž 5.5. and Ž 5.4. into Ž 5.6. we have the tangent gyroline yž t. Ž a b t. gyra b t, ab t Ž 5.7. Ž. 0 0 of the gyroline vž. t, Ž 5.5., at the point t 0. With several gyrovector space algebraic manipulations we simplify it to the point where it is recognized as the gyroline vž. t itself but with a new parameter. We thus have yž t. Ž a b t. Ž gyra b t, ab. t Ž. 0 0 Ž a b t. gyra b t, až b t. Ž Ž a b t. gyra b t, b t Ž b t. Ž Ž a b t. gyra, b t Ž b t. Ž 4. Ž a Ž b t0 b t. a b Ž t t. Ž recovering the gyroline vž. t parametrized by a new parameter, t0 t, replacing the old parameter t, Ž The derivation of each of the equations in the chain of Eqs. Ž 5.8. follows. Ž. follows from the gyroautomorphism property Ž V8.. Ž. 2 follows from a right loop Ž G9. followed by a left cancellation Ž G0..

13 THE POINCARE BALL MODEL 333 Ž. 3 follows from a left loop Ž G5.. Ž. 4 follows from the left gyroassociative law. Ž. 5 follows from the scalar distributive law Ž V2.. REFERENCES. L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGrawHill, New York, H. Haruki and T. M. Rassias, A new characteristic of Mobius transformations by use of Apollonius quadrilaterals, Proc. Amer. Math. Soc. 26 Ž 998., S. G. Krantz, Complex Analysis: The Geometric Viewpoint, Math. Assoc. of America, Washington, DC, G. Saad and M. J. Thomsen Ž Eds.., Nearrings, Nearfields and K-Loops, Kluwer Academic, Dordrecht, A. A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. Ž 988., A. A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, Resultate Math. 6 Ž 989., A. A. Ungar, Thomas precession and its associated grouplike structure, Amer. J. Phys. 59 Ž 99., A. A. Ungar, Extension of the unit disk gyrogroup into the unit ball of any real inner product space, J. Math. Anal. Appl. 202 Ž 996., A. A. Ungar, Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 Ž 997., A. A. Ungar, From Pythagoras to Einstein: The hyperbolic Pythagorean theorem, Found. Phys. 28 Ž 998., A. A. Ungar, The hyperbolic Pythagorean theorem in the Poincare disc model of hyperbolic geometry, Amer. Math. Monthly 06 Ž 999., A. A. Ungar, The bifurcation approach to hyperbolic geometry, Found. Phys. 303 Ž 2000., A. A. Ungar, Gyrovector spaces in the service of hyperbolic geometry, in Mathematical Analysis and Applications Ž T. M. Rassias, Ed.., pp , Hadronic Press, FL, A. A. Ungar, Hyperbolic trigonometry and its application in the Poincare ball model of hyperbolic geometry, Comput. Math. Appl., in press. 5. A. A. Ungar, Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry, Comput. Math. Appl. 40 Ž 2000., A. A. Ungar, Mobius transformations of the ball, ahlfors rotation and gyrogroups, in Nonlinear Analysis in Geometry and Topology Ž T. M. Rassias, Ed.., pp , Hadronic Press, FL, A. A. Ungar, The relativistic composite-velocity reciprocity principle, Found. Phys. 30 Ž 2000.,