DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

92 Chapter 3. Surfaces: Further Topics*4. Using Proposition 1.2, prove that horizontal lines have constant geodesic curvature κ g =1inH; prove that this is also true for circles tangent to the u-axis. (These curves are classicallycalled horocycles.)5. Prove that every curve in H of constant geodesic curvature κ g =1is of the form given inExercise 4. (Hints: Assume we start with an arclength parametrization (u(s),v(s)), and useProposition 1.2 to show that we have 1 = u′v + θ′ and u ′2 + v ′2 = v 2 . Obtain the differentialequation(v d2 v( dv) ) 2 3/2 ( dv) 2du 2 = 1+− − 1,duduand solve this by substituting z = dv/du and getting a separable differential equation fordz/dv.)6. Let T a,b,c,d (z) = az + b , a, b, c, d ∈ R, with ad − bc =1.cz + da. Suppose a ′ ,b ′ ,c ′ ,d ′ ∈ R and a ′ d ′ − b ′ c ′ =1. Check thatT a ′ ,b ′ ,c ′ ,d ′◦ T a,b,c,d = T a ′ a+b ′ c,a ′ b+b ′ d,c ′ a+d ′ c,c ′ b+d ′ d(a ′ a + b ′ c)(c ′ b + d ′ d) − (a ′ b + b ′ d)(c ′ a + d ′ c)=1.Show, moreover, that T d,−b,−c,a = T −1a,b,c,d . (Note that T a,b,c,d = T −a,−b,−c,−d . The readerwho’s taken group theory will recognize that we’re defining an isomorphism between thegroup of linear fractional transformations and the group SL(2, R)/{±I} of 2 × 2 matriceswith determinant 1, identifying a matrix and its additive inverse.)b. Let T = T a,b,c,d . Prove that if z = u + iv and v>0, then T (z) =x + iy with y>0.Deduce that T maps H to itself bijectively.7. Show that reflection across the geodesic u =0isgiven by r(z) =−z. Use this to determinethe form of the reflection across a general geodesic.8. The geodesic circle of radius R centered at P is the set of points Q so that d(P, Q) =R. Provethat geodesic circles in H are Euclidean circles. One way to proceed is as follows: The geodesiccircle centered at P =(0, 1) with radius R =lna must pass through (0,a) and (0, 1/a), andhence ought to be a Euclidean circle centered at (0, 1 2(a +1/a)). Check that all the points onthis circle are in fact a hyperbolic distance R away from P . (Hint: It is probably easiest towork with the cartesian equation of the circle.)Remark. This result can be established rather painlessly by using the Poincaré disk modelgiven in Exercise 15.and*9. What is the geodesic curvature of a geodesic circle of radius R in H? (See Exercises 4 and 8.)10. Recall (see, for example, p. 298 and pp. 350–1 of Shifrin’s Abstract Algebra: A GeometricApproach) that the cross ratio of four numbers A, B, P, Q ∈ C ∪ {∞} is defined to be[A : B : P : Q] = Q − A / Q − BP − A P − B .