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）︰2013/11/1 考試時限（分鐘）：170 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : (15pt) 1. Prove the iso-perimetric inequality l≧4πA for piecewise C^1 simple closed plane curves, and the equality holds if and only if the curve is a circle. (20pt) 2. Let α:I→R^3 be a regular curve parametrized by arc length with 0∈I,κ>0. (a) Derive the local cononical form of α(s) at α(0)in terms of the Frenet frame at s=0, and sketch the projection curves on the TN, TB, NB planes respectively. (b) Assume that κ,κ',τ≠0 and denote by R=1/κ,T=1/τ. Show that α lies in a sphere of radius r iff R^2 + (R'T)^2 = r^2. (20pt) 3. Let α:[0,l]→R^3 be a regular curve parametrized by arc length so that the Frenet frame is denoted, and let S be the tubular surface along α of radius r>0. (a) Determine the range of r so that S id locally a regular surface, (i.e. an immersion). (b) Compute its area. (c) If α is moreover simple (injective), show that S is a regular surface for small r. (20pt) 4. Let N:S→S^2 be the Gauss map with (U,x) a coordinate chart for p∈S. (a) Show that the matrix for dN_p with respect to the bases x_u, x_v of T_p S is given by ( fF-eG gF-fG ) 1 / (EG-F^2) * ( eF-fE fF-gE ) //一個2*2矩陣 (b) Derive the differential equation for lines of curvature, and show that the coordinate curves are lines of curvature if and only if F = f = 0. (c) Derive the equation for asymptotic curves. Solve them for Enneper's surface x(u,v) = (u - u^3 / 3 + uv^2, v - v^3 / 3 + vu^2, u^2 - v^2). (20pt) 5. Let S be a surface of revolution on a curve α in the xz plane. (a) For α being parametrized by arc length s, compute E,F,G,e,f,g and K,H. (b) Determine all such regular surfaces S with K=1. When is S compact? (c) Determine all minimal surfaces of revolution. (15pt) 6. Let S be the graph defined by z=f(x,y) over a compact domain D⊂R^2, with ∂D a smooth curve. Let h be C^∞ on D with h|_∂D = 0. Consider variations of surfaces S_t defined by z=f+th with A(t) being its area. (a) Compute H for S, and show that A'(0)=0 for all h if and only if H=0. (b) For S a minimal graph, show that A''(0)>0 for any h≠0. (10pt) 7. Let S be a minimal surface. Construct isothermal coordinates near any non- planar points. (Show first that <dN_p(v),dN_p(w)>=-K(p)<v,w>.) --

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