[試題] 102-1 王金龍 幾何學 期末考
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
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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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※ 編輯: craig100 來自: 114.44.132.94 (01/08 12:29)