課程名稱︰幾何學 課程性質︰數學系大三必修 課程教師︰蔡宜洵 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015/1/17 考試時限（分鐘）：110 試題 : Total points: 105 Notice: any theorems or facts put to use in the process of your proof or computation must be stated clearly. 1 (20 pts in total). i) (15 pts) Given two (smooth) simple closed curves C_1, C_2 on a sueface S of Gaussian curvature K < 0, such that they enclose a region diffeomorphic to a cylinder. Suppose C_1 is a geodesic. Show that C_2 cannot be geodesic. ii) (5 pts) Let S be a compact surface without boundary with K > 0 and C_1 be a simple closed geodesic which encloses a region R of S. Suppose C_2, which belongs to R, is a smooth geodesic which divides R into two connectes sets D_1, D_2 and is orthogonal to C_1 at the two points of intersection. Show that the integral of the curvature ∫∫ K dσ = π, i=1,2. D_i 2 (20 pts in total.) Let T be a torus of revolution parametrized by x(u,v) = ((rcosu + a)cosv,(rcosu + a)sinv,rsinu). i) (15 pts) If a geodesic C is tangent to the parallel u= π/2, then it is entirely contained in the region given by -π/2 ≦ u ≦ π/2. ii) (5 pts) Is it true that C must intersect the maximal parallel u = 0? Prove your answer. 3 (20 pts in total). Let S be a surface obtained by revolving the tractrix z = g(x) with g'(x) =√1-x^2/x, g(1) = 0. i) (10 pts) Compute the Christoffel symbols — k ∣ ij, 1 ≦ i,j,k ≦ 2. ii) (10 pts) Given a point p on S. Find a local parametrization in u,v such that the first fundamental form around p is given by 2 2 2 ds = du + G(u,v)dv, for an explicit G(u,v). Show the geometric meaning of these coordinates u,v. 2 2 2 4 (15 pts in total). Suppose ds = du + G(u,v)dv on S, with p = (0,0) and √G(u,v)| = 1, √G (u,v)| = 0. Let a square Q_a be centered at p with u=0 u u=0 vertices at (a,a), (-a,a), (-a,-a) and (a,-a). i) (10 pts) Show that around p, 2 √G = 1 - K(p)*u /2 modulo higher order terms in u,v. ii) (5 pts) Show that K(p) = 8a-L(Q_a) lim ___________ , a→0 3 2a where L(Q_a) is the perimeter of Q_a. 5 (15 pts in total). Let C be a space curve α(t), t arc length parameter, with k(t) =/= 0 for all t. Construct the tangent surface S as a ruled surface with the rulings given by tangent lines to C. i) (8 pts) Given a 2 parametrization of S and find the first fundamental form ds of S (where it is smooth). ii) (7 pts) Can it be (locally) isometric to a plane? Prove your answer. 6 (15 pts in total). Let the unit sphere be parametrized by (sinψcosθ, sinψsinθ,cosψ). Let C be a curve and w_p be a unit vector tangent to C at some point p of C. For the parallel transport of w_p along C compute the angle between the tangent ventor t_q and w_q at q in C in the following cases. i) (10 pts) C is given by ψ = ψ_0, θ_1 ≦ θ ≦θ_2, with p at θ= θ_1 and q at θ = θ_2. ii) (5 pts) C is given by (the equation) ψ = θ, θ_1 ≦ θ ≦θ_2, with p at θ= θ_1 and q at θ = θ_2. (hint: you may use Dw 1 dφ [—] = ————{G v' - E u'} + — for ii).) dt 2√EG u v dt --

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