課程名稱︰幾何學
課程性質︰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）︰2010.1.14.
考試時限（分鐘）：18:10 ~ 21:10
是否需發放獎勵金：是！
試題 :
Important: Give your solutions in detail. Each problem deserves 20 points.
1. Denote by S_t, t in (-ε,ε), a normal variation of S = x(U) defined by
t
x = x + thN for ome smooth function h, and let A(t) be he area of S_t.
(1) Show that S has H≡0 (minimal surface) if and only if A'(0)= 0
for any such S_t.
(2) For S being a minimal surface, show that
〈dN_p(w_1) , dN_p(w_2) 〉= —K(p)〈w_1, w_2〉for any w_1,w_2 in T_p(S).
2. Define the notion of geodesics on a regular surface and derive the
differential equations of the geodesics α(t)= x(u(t),v(t)).
For a surface of revolution x(u,v)=(f(v)cosu, f(v)sinv, g(v)), prove that
fcosθ takes constant value along geodesics, whereθis the angle between
x_u and α'(t).
3. Use the Gauss-Bonnet theorem to Prove Jacobi's theorem:
If a closed regular curve in R^3 has k > 0 and its principal normal n(s)
form a curve γ on S^2 without self-intersections, thenγ seperates S^2
into two regions with equal area.
4. Use the Gauss-Bonnet theorem to show that
(1) Let S be a regular surface such that the parallel transport between
any two points in it is independent of the path, then K = 0 on S.
(2) Let S be a regular surface homeomorphic to a cylinder with K < 0,
then S has at most one simple closed geodesic.
5. Use the geodesic polar coordinates to show that
(1) Any two surfaces with the same constant curvature K are locally
isometric.
(2) Let A(r) be the area of the geodesic ball of radius r centered at
p in S, then
12 πr^2 － A(r)
K(p) = lim ─ ──────── .
r→0 π r^4
6. State and prove the Gauss-Bonnet theorem.
※ 編輯: iamwjy 來自: 125.226.128.200 (01/16 14:11)