課程名稱︰幾何學 課程性質︰數學系選修 可抵必修幾何學導論 課程教師︰王金龍教授 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰105.10.21 考試時限（分鐘）：3小時 40分 試題 : 1. (a)(10 pts) Prove that a curve in R^3 lies on a sphere of radius R if and only if its curvature k and torsion τ satisfy 1 (dk/dl)^2 R^2 = --- ( 1 + --------- ). k^2 (τk)^2 (b)(10 pts) For a curve γ in R^3 with general parameter, show that |[γ',γ'']| -(γ',γ'',γ''') k = -------------, τ = ------------------. |γ'|^3 |[γ',γ'']|^2 2. Let γ: [0,L] → R^n be a C^∞ curve parametrized by arc length l in [0,L]. Assume that γ is n-1 regular in the sense that γ'(l), γ''(l), ..., γ^(n-1)(l) ≠ 0 for any l. (a)(10 pts) Show that a Frenet unitary frame T = (t1,...,tn)^t can be defined so that t1 = γ' and | k1 | |-k1 k2 | T' = | -k2 ... | T, | ... kn-1| | -kn-1 | where all the other entries are zero. (b)(10 pts) Conversely, given k1,...,kn-1 in C^∞([0,L]), show that up to Euclidean motion there is a unique curve γ so that its Serret-Frenet frame takes the above form. 3. Let S in R^3 be a non-singular surface with parametrization γ(u,v), and let N: S → S^2 be its Gauss map. Denote the two fundamental forms by dl2 = Edu^2 + 2Fdudv + Gdv^2 and Ldu^2 + 2Mdudv + Ndv^2. (a)(5 pts) If the two principle curvatures of a connected surface S coincide λ1 = λ2 = λ at all points, show that λ is a constant and S is a portion of a sphere or a plane. (b)(7 pts) Show that the matrix for dNp, p in S, in the basis γ1, γ2 of TpS is given by 1 | MF - LG LF - ME | ------| |. EG-F^2| NF - MG MF - NE | Deduce that the Gaussian curvature K equals (LN-M^2)/(EG-F^2) (c)(8 pts) Gauss' famous Theorema Egregium says that K depends only on the first fundamental form dl^2. Prove it by any method you know. For example, one way to do it is to show that when F = 0, K can be written in terms of A = √E and B = √G as -1 A_2 B_1 K = --( (---)_2 + (---) ). AB B A Anotehr way is to prove it under isothermal coordinates. 4. Consider the surface S given by the graph of z = f(x,y) over R^2. (a)(10 pts) Show that the mean curvature is given by ▽f H = div(-------------). √(1+|▽f|^2) (b)(5 pts) Let B in R^2 be an open ball, h in C^∞(B's closure), h is 0 on the boundary of B, and Ah(t) be the area of the surface St defined by z = f(x,y) + th(x,y). Show that Ah'(0) = 0 for all such h if and only if H = 0 over B. (c)(5 pts) If H = 0 over B, is that true S has minimal area among all St for t small? 5. Three models of two dimensional hyperbolic geometry. (a)(4 pts) Lobachevsky plane: Let L^2 in R^{1,2} be the upper half of the space-like surface t^2 - x^2 - y^2 = 1, and dl^2 be the positive metric induced from R^{1,2}. Show that dl^2 = dχ^2 + sinh^2χ dφ^2 in the pseudo-spherical coordinates (χ,φ). (b)(4 pts) Poincare model: Compute the stereographic projection L^2 → D^2 from the south pole (0,0,-1) onto the unit disk D^2, and show that the induced metric is given by 4|dw|^2 dl^2 = -------------. (1 - |w|^2)^2 (c)(4 pts) Klein model: Construst a Mobius transformation M: H～D^2, where H = {z = x + yi in C | y > 0}, and show the induced metric is given by dx^2 + dy^2 dl^2 = -----------. y^2 (d)(4 pts) Determine the (direct) isometry group in one of the above three models. (e)(4 pts) Determine the set of all geodesics in one of the three models of hyperbolic plane. In particular show that given a point p not in a geodesic γ there are infinitely many geodesics passing through p and disjoint from γ. 6. Theorem L: A surface S with dl^2 = e^φ|dz|^2 and constant K is locally isometric to an open subset of either S_R^2, R^2 or L_R^2. Prove it by the following steps. Let ψ(z) := φzz - (1/2)φz^2. (a)(4 pts) Show that ψ is analytic. (b)(4 pts) Under an analytic coordinate change z = f(w) we get dl^2 = e^φ~|dw|^2 and the corresponding ψ~(w). Show that f''' 3 f'' φ~(w) = φ(z) + log|f'(w)|^2, ψ~(w) = (f'(w))^2ψ(z) + (---- - -(-----)^2)(w). f' 2 f' (c)(4 pts) Denote the above RHS term (the Schwarzian derivative) by S(z;w) := S(f;w). Show that S(z;w) = -S(w;z)(f')^2, and for any two independent solutions g1, g2 of the ODE g''(z) + I(z)g(z) = 0 we have S(g1/g2;z) = 2I(z). (d)(4 pts) Show that there exists analytic f such that ψ~ = 0 in (b). Then conclude that e^(-ψ~/2) = a|w|^2 + bw + b_bar w_bar + c for some a,c in R and b in C. (e)(4 pts) Complete the proof of Theorem L by a further change of coordinate. -- 正妹也不過就是一組物質波方程式的特解罷了 --

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