DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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88 Chapter 3. Surfaces: Further Topicsand Q lie on a vertical line, then the vertical ray through them is the unique geodesic joining them.If P and Q do not lie on a vertical line, let C be the intersection of the perpendicular bisector ofPQ and the u-axis; then the semicircle centered at C is the unique geodesic joining P and Q.Example 1. Given P, Q ∈ H, wewould like to find a formula for the (geodesic) distanced(P, Q) between them. Let’s start with P =(u 0 ,a) and Q =(u 0 ,b), with 0
§2. An Introduction to Hyperbolic Geometry 89[ ] a b(i) Composition of functions corresponds to multiplication of the 2 × 2 matriceswith determinant 1, so we obtain a group.(ii) T maps H bijectively to H.(iii) T is an isometry of H.We leave it to the reader to check the first two in Exercise 6, and we check the third here. Giventhe point z = u + iv, wewant to compute the lengths of the vectors T u and T v at the image pointT (z) =x + iy and see that the two vectors are orthogonal. Note thatso y =so we have(a(u + iv)+b)(c(u − iv)+d)az + b (az + b)(cz + d)=cz + d |cz + d| 2 =|cz + d| 2(ac(u 2 + v 2 )+(ad + bc)u ) + i ( (ad − bc)v )=v|cz + d| 2 .Nowwehave3and, similarly, ˜G =x 2 v + y 2 vy 2x u + iy u = −ix v + y v = T ′ (z) =|cz + d| 2 ,(cz + d)a − (az + b)c(cz + d) 2 =Ẽ = x2 u + yu2y 2 = 1 y 2 |T ′ (z)| 2 = 1 y 2 · 1|cz + d| 4 = 1 v 2 = E,= G. Onthe other hand,˜F = x uy u + x v y vy 2 = x u(−x v )+x v (x u )y 2 =0=F,1(cz + d) 2 ,as desired.Now, as we verify in Exercise 10 or in Exercise 12, linear fractional transformations carrylines and circles in C to either lines or circles. Since our particular linear fractional transformationspreserve the real axis (∪{∞}) and preserve angles as well, it follows that vertical lines and semicirclescentered on the real axis map to one another. Thus, our isometries do in fact map geodesics togeodesics (how comforting!).If we think of H as modeling non-Euclidean geometry, with lines in our geometry being thegeodesics, note that given any line l and point P /∈ l, there are infinitely many lines passingthrough P “parallel” to (i.e., not intersecting) l. Aswesee in Figure 2.3, there are also two specialcdlPFigure 2.3lines through P that “meet l at infinity”; these are called ultraparallels.3 These are the Cauchy-Riemann equations from basic complex analysis.
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DIFFERENTIALGEOMETRY:A First Course
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4 Chapter 1. Curves2πb2πaFigure 1
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6 Chapter 1. CurvesAlternatively, s
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8 Chapter 1. CurvesAn important obs
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10 Chapter 1. Curvesof the road, st
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12 Chapter 1. CurvesSummarizing,κ(
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14 Chapter 1. Curvescurvature κ. I
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16 Chapter 1. CurvesFigure 2.2Conve
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18 Chapter 1. Curvesalso Exercise 2
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20 Chapter 1. Curvesto the curve β
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22 Chapter 1. Curvesthe osculating
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24 Chapter 1. CurvesWe turn now to
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26 Chapter 1. Curvesintersect C eit
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28 Chapter 1. Curvesh(s,t)α(t)α(s
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30 Chapter 1. Curvesy( x(s) ,y(s))
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32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFig
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CHAPTER 2Surfaces: Local Theory1. P
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36 Chapter 2. Surfaces: Local Theor
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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