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)