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GEODESICS OF HYPERBOLIC SPACE WILL ADKISSON Abstract. Hyperbolic geometry is a non-Euclidean geometry in which the traditional Euclidean parallel postulate is false. Instead an alternate version holds; namely that given a point and a line, there exist at least two lines parallel to the first passing through the point. The purpose of this paper is to define the geodesics of the hyperbolic plane. This is accomplished by showing that the set of all isometries of the hyperbolic plane is in fact equivalent to the set of M¨obiustransformations that are bijective on the space. For the upper half-plane model of hyperbolic geometry, this set is PSL(2,R). We show that the line through the y-axis of the upper half plane is a geodesic, and examine the potential results when this line is transformed by M¨obiustransformations to determine the geodesics of the space. We can then verify that the parallel postulate is in fact false in the upper half-plane and show that this alternate version holds. Contents 1. Introduction to Hyperbolic Geometry 1 2. M¨obiustransformations 2 3. SL(2,R) and PSL(2,R) 3 4. Isometries of Hyperbolic Space 6 5. Geodesics in Hyperbolic Space 9 6. Parallel Lines in Hyperbolic Space 13 Acknowledgments 14 References 14 1. Introduction to Hyperbolic Geometry Hyperbolic geometry cannot be isometrically embedded in Euclidean space, so several Euclidean models have been constructed to examine specific aspects of hy- perbolic geometry. In this paper, we will deal with two models of hyperbolic geom- etry, the Poincar´edisc model and the upper half-plane model. Definition 1.1. The disc model of hyperbolic space, D, consists of the unit disc in the complex plane, that is, the set U = fz = x + iyjpx2 + y2 < 1g. The metric 2 2 2 4(dx +dy ) dzdz of D is ds = (1−x2+y2)2 = (1−|zj2)2 . Definition 1.2. The upper half-plane model of hyperbolic space, H, consists of the upper half of the complex plane, not including the real line; that is, the set 2 dx2+dy2 H = fz = x + iyjy > 0g. The metric of H is ds = y2 1 2 WILL ADKISSON The disk and half-plane models of hyperbolic space are isomorphic, mapped conformally by the transformation w = eiθ z−z0 , where θ is a constant value. Note z−z0 that the real line on the edge of H maps to the edge of the disc in D. It can be helpful to extend the upper half plane model to the upper half of the Riemann sphere. The Riemann sphere is the complex plane joined with a point at infinity. All vertical lines on the complex plane are considered to intersect at this point. This set C [ f1g can be seen as a sphere, with 0 and 1 at opposite poles. This allows for simple definitions of otherwise indeterminant functions, such as division by zero. z z Definition 1.3. For every z 6= 0 We define 0 = 1 and 1 = 0 on the Riemann sphere. 2. Mobius¨ transformations A class of especially well-behaved functions in hyperbolic space is the set of M¨obiustransformations. Definition 2.1. M¨obiustransformations on the upper half plane are functions of az+b the form f(z) = cz+d , where a; b; c; d 2 C and ad − bc = 1. Because the quantity ad − bc is suspiciously similar to the determinant of a 2 × 2 matrix, it is referred to as the `determinant` of the M¨obiustransformation. M¨obiustransformations are relatively simple transformations, and can be seen as the composition of even simpler functions. az+b a b Lemma 2.2. Let m(z) = cz+d . If c = 0, then m(z) = d z + d . If c 6= 0 then 2 1 a m(z) = f(i(g(z))), where g(z) = c z + cd, i(z) = z , and f(z) = −z + c . az+b a b Proof. Suppose c = 0. Then m(z) = d = d z + d Suppose c 6= 0. Then az + b (az + b)c acz + bc m(z) = = = cz + d (cz + d)c c2z + cd Remember that ad − bc = 1. Thus acz + bc acz + ad − (ad − bc) a 1 m(z) = = = − = f(i(g(z))) c2z + cd c2z + cd c c2z + cd This result can be generalized even further, to get the following description of M¨obiustransformations: Theorem 2.3. M¨obiustransformations consist of compositions of the following types of maps: • Scalings- maps of the form z 7! kz for some k 2 C • Translations - maps of the form z 7! z + k for some k 2 C. 1 • Inversions- maps of the form z 7! z Proof. As shown above, any M¨obiustransformation is a composition of the func- 1 2 tions f(z) = az + b, i(z) = z , and g(z) = c z + cd. f is the composition of scalings and translations, namely f1(z) = az and f2(z) = z + b. i(z) is an inversion. g(z) is also the composition of scalings and translations, namely g1(z) = c2z and g2(z) = z + cd. GEODESICS OF HYPERBOLIC SPACE 3 We will now examine some general properties of M¨obiustransformations. Theorem 2.4. The set of M¨obiustransformations is closed under composition. a1z+b1 Proof. Let f1 and f2 be M¨obiustransformations. Then f1(z) = and f2(z) = c1z+d1 a2z+b2 . c2z+d2 a2z+b2 a1( ) + b1 c2z+d2 (f1 ◦ f2)(z) = f1(f2(z)) = a2z+b2 c1( ) + d2 c2z+d2 a1a2z+b2+b1c2z+b1d2 c2z+d2 (a1a2 + b1c2)z + (b1d2 + b2) = c a z+b +d c z+d d = 1 2 2 1 2 1 2 (c1a2 + d1c2)z + (d1d2 + b2) c2z+d2 This is clearly another M¨obiustransformation. Theorem 2.5. The inverse of a M¨obiustransformation is another M¨obiustrans- formation. az+b dz−b Proof. Let f be a M¨obiustransformation. Thus f = cz+d . Let g = −cz+a . Clearly, g is a M¨obiustransformation. az+b daz+bd−bcz−bd d cz+d − b daz − bcz g(f(z)) = = cz+d = = z az−b −caz−bc+acz+ad da − bc −c cz−d + a cz+d Thus g = f −1, so f −1 is a M¨obiustransformation. 3. SL(2,R) and PSL(2,R) Not every M¨obiustransformation is defined from H to H. We will now examine the subset of M¨obiustransformations that are. To classify these transformations it is helpful to look at the M¨obius transformations acting on the upper half plane as the action of a group of matrices on the upper half plane. The properties of these matrices are illustrative of the action of the transformations, and make them easier to classify. a b Definition 3.1. SL(2 ) = such that a; b; c; d 2 and ad − bc = 1 R c d R Definition 3.2. P SL(2R) = SL(2R)={±1g Theorem 3.3. The projective action of P SL(2; R) on the set of lines through the origin in C2 is equivalent to the action of M¨obiustransformations on the complex plane. a b Proof. Let A 2 P SL(2 ). Thus A = . Let c be a line through the origin R c d v in C2, expressed as a vector. Thus c = . Because the action of P SL(2 ) is w R v 0 projective, we multiply this vector by the scalar 1 to get w : Defining z = v w 1 w z gives the vector : Multiplying these matrices together gives 1 4 WILL ADKISSON a b z az + b = : c d 1 cz + d az+b Using the same technique as before, this is equivalent to cz+d : Thus ele- 1 z az+b ments of P SL(2 ) map 7! cz+d , which is exactly the action of M¨obius R 1 1 transformations with real coefficients. As we have shown, M¨obiustransformations are compositions of translations, scales, and inversions. To find the translations that act on the upper half plane, we will look at each in turn. Translations of the form z 7! z + k will be defined on H if and only if they do not translate downwards; that is, if k is a real number or k is a positive complex 1z+b number. Thus if the translation is equal to 0z+1 , then b is real or b is a positive complex number. Scales (z 7! kz) will only be defined on H if k is a positive real number. If k 2 R ≤ 0 then a point in H will be mapped to the real line (if k = 0) or to a point with a negative imaginary value, neither of which are part of H. If k 2 C n R, then k = a0 + b0i. Thus a + bi 7! aa0 + (ab0 + a0b)i − bb0. Thus when ab0 + a0b ≤ 0, this is undefined. Solving this inequality reveals a set of points in H for which this operation maps outside the upper half plane. 1 The inversion, z 7! z , isn't defined on H. This can be seen quite easily; let z = i. 1 i 7! i = −i, which is not in the upper half plane. To get around this, however, we k look at the composition of inversions with a scalar: z 7! z . By the same reasoning, k k cannot be positive, but when k is negative, the function is defined. i 7! i = −ki, which is in the upper half plane. A pretty obvious candidate for the M¨obiustransformations on H is the set of M¨obiustransformations with all real coefficients.
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