課程名稱︰幾何學 課程性質︰數學系必修 課程教師︰王金龍 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2014/1/3 考試時限（分鐘）：180mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : ∂x ∂x ( Notation: ── denotes x_1 , and ── denotes x_2. ) ∂u ∂v There are 2 pages with 6 problems. Each problem deserves 20 points. Show your answers/computations/proofs in details. You may work on each part independently. 1.Let S in R^3 be a parametrized rugular surface difined by x(u,v). (a)Derive the Codazzi equations: 1 2 1 2 e - f = eΓ + f(Γ - Γ ) - gΓ 2 1 12 12 11 11 2 1 2 1 g - f = gΓ + f(Γ - Γ ) - eΓ 1 2 12 12 22 22 (b)If the coordinate curves are lines of curvature , simplify the equations to E_2 e g G_1 e g e = ── ( ─ + ─ ) , g = ── ( ─ + ─ ) 2 2 E G 1 2 E G (c)Is there a surface with E = G = 1 , F = 0 and e = 1 , g = -1 , f = 0 ? How about E = 1 , F = 0 , G = (cosu)^2 and e = (cosu)^2 , f = 0 g = 1 ? You may use Bonnet's theorem , and the foumula fou K without proving it that fou F = 0, -1 E_2 G_1 K = ──── ( ( ──── ) + ( ──── ) ) 2√(EG) √(EG) 2 √(EG) 1 2.Let γ(t) = x(u(t),v(t)) be a curve on S (a)Define the notion for γ to be a geodesic and derive its equations in terms of local coordinates (u(t),v(t)). (b)If S is a Liouville surface , namely E = G = U(u) +V(v) , F = 0 . Show that any geodesic γ satisfies U(sinθ)^2-V(cosθ)^2 = c where θ=∠(γ',x_1) and c is a constant. 3. (a)Compute ind v at p = (0,0) in the following case: p (i) v(x,y) = ( x^2 - y^2 , -2xy ) (ii) v(x,y) = ( x^3 - 3xy^2 , y^3 - 3x^2y ) (b)Can it happen that ind v = 0 for p a singular point of v? p If so , give an example. (c)Let C in S^2 be a regular closed curve , v a vector field on S whose trajectories are never tangent to C. Prove that each region R with ∂R = C contains some singular point of v. 4.Using the geodesic polar coordinates to prove: (a)Any two surfaces with the same constant curvature K are locally isometric. (b)Let A(r) be the area of the geodesic ball B (p) , then r 12 πr^2 - A(r) K(p) = ─ lim ( ────── ) π r→0 r^4 5.(Poincare models for hyperbolic geometry) 2 |dw|^2 Let |H = { w | Im(w) > 0 } with ds = ──── (Im(w))^2 2 4|dz|^2 and |D = { z | |z| < 1 } with ds = ────── (1-|z|^2)^2 w - i (a)Show thar |H → |D , w|→z = ─── is an isometry. w + i (b)Determine all geodesics in |D (c)Let Ω be the region bounded by the 4 unit circles cintered at (±1 , ±1) 4dxdy Compute ∫ ───────── Ω (1 - x^2 - y^2)^2 ( You may use Gauss-Bonnet theorem or do it directly. ) 6.State and prove the Gauss-Bonnet theorem. ( A very good solution to this probelm may get some extra credits. ) * If you prefer to write proofs , you may replace up to 2 problems from 1 to 5 , but not 6 , by ( stating and proving ) the following: (Ⅰ)Any compact S in R^3 with constant K must be a sphere. π (Ⅱ)2nd variation formala for normal variations and Bonnet's theorem on d ≦ ─ √k (Ⅲ)Fenchel's theorem and Fary-Milnor's theorem on ∫kds < 4π => unknotted. (Ⅳ)Hilbert's theorem on complete surface S with K = -1 . (Ⅴ)Something important you had well prepared but not listed above. Label your solution by n* if it is for the n-th problem. Notice that you will still get at most 20 points for that problem. 註:就是你可以選擇*區的題目 和你考卷上的某個小題對調 就不用寫你排除的考題 但最多只能對調兩個小題 且第6題不可和*區對調 -- 每題一分 1.會上BBS 5.會上表特版 9.知道星野亞希 13.被正妹發過卡 2.1週1碗泡麵↑ 6.經常說:「宅」 10.讀理工學院或資訊科系14.意淫女主播 3.會上黑特版 7.常上西斯版 11.有5566或183帳號 15.下載/續傳軟體正開著 4.玩過魔獸 8.情人節沒事幹 12.幫修過電腦 16.收藏模型 0 不宅 4 宅 8 很宅 12 超宅 16 --

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