课程名称︰几何学 课程性质︰数学系大三必修 课程教师︰崔茂培 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2015/11/13 考试时限（分钟）：165 试题 : Total = 110 points. Enjoy the exam on black Friday!!! 2 (1) (20 pts) A regular spherical curve γ(s): I → S (1) parametrized by arc- length has an alternate set of equations that describe its properties. Note that γ(s) is also normal to the sphere, then the signed normal can be defined as the vector S = γ ×T. Then {T, S, γ} becomes an orthonormal frames along γ(s). (a) (6 pts) Show that dT ─ = κ S - γ ds g dS ─ = -κ T ds g dγ ──= T ds dT where the geodesic curvature κ is defined as κ = ─．S. g g ds (b) (10 pts) Show that dκ 2 2 1 g κ = 1 + κ , τ = ────── g 2 ds 1 + κ g 1 1 N = ─(κ S - γ), B = ─(κ γ + S). κ g κ g (c) (4 pts) Show that γ is planar if and only if the curvature is constant. (2) (15 pts) For a regular space curve γ(s) parametrized by arc-length, we say dX that a normal field X is parallel along γ if X．T = 0 and ─ is parallel to ds T. (a) (3 pts) Show that for a fixed s and X(s ) ⊥ T(s ) there is a unique 0 0 0 parallel field X that is X(s ) at s . 0 0 (b) (6 pts) A Bishop frame consists of an orthonormal frame T, N , N along 1 2 the curve so that N , N are both parallel along γ. For such a frame 1 2 show that dT ─ = κ N + κ N ds 1 1 2 2 dN 1 ──= -κ T ds 1 dN 2 ──= -κ T ds 2 Note that such frames always exist, even when the space curve doesn't have positive curvature everywhere. 2 2 2 (c) (3 pts) Show further that for such a frame κ = κ + κ . 1 2 (d) (3 pts) Show that if γ has positive curvature so that N is well- dφ defined, then N = cosφ N + sinφ N , B = -sinφ N + cosφ N and ── 1 2 1 2 ds = τ where φ(s) is defined by κ = κcosφ and κ = κsinφ. 1 2 (3) (15 pts) Recall that the first fundamental of a regular surface σ can be ╭ σ ．σ σ ．σ ╮ │ u u u v │ identified with the matrix Ⅰ = │ │, the second │ σ ．σ σ ．σ │ ╰ u v v v ╯ fundamental can be identified with the matrix Ⅱ = ╭ -σ ．N -σ ．N ╮ │ u u u v │ │ │. We can define the third fundamental as the first │ -σ ．N -σ ．N │ ╰ v u v v ╯ ╭ N ．N N ．N ╮ │ u u u v │ fundamental form of the Gauss map N, i.e., Ⅲ = │ │. Prove │ N ．N N ．N │ ╰ u v v v ╯ that Ⅲ - 2HⅡ + KⅠ = 0. (4) (10 pts) If the first fundamental form of the surface is I = 2 2 -1 2 2 (1 + u + v ) (du + dv ), compute the Gaussian curvature of the surface. (5) (20 pts) Let C be a circle of radius 10 that contains the points (0, 0, 8) and (0, 0, -8), and let A be the (open) minor arc of C between these points. Let S be the surface obtained by rotating the arc A around the z-axis, oriented so that normal vectors point outwards. http://i.imgur.com/F11kUGK.png Note that the surface S does not include the cusp point (0, 0, 8) and (0, 0, -8). (a) Find the principal curvatures of S at the point (4, 0, 0). (b) Find the image of S under the Gauss map. Express your answer as one or more inequalities defining a region on the unit sphere. (c) Use your answer to part (b) to evaluate ∫ K dA, where K is the Gaussian S curvature of S. (Use geometry to find the answer.) 3 (6) (10 pts) Suppose that M is a compact orientable surface in |R and K > 0 everywhere on M. (a) What can you say about the topology of M and why? (b) Show that the Gauss map is a one-to-one and onto map. 3 (7) (10 pts) If M is a compact orientable surface in |R and has constant Gaussian curvature, then it is a round sphere. (8) (10 pts) Suppose that a curve γ lies in two surfaces S and S that make a 1 2 constant angle along γ (i.e. their tangent plane make a constant angle). Show that α is principal in S (i.e. α'(t) is a principal direction for S 1 1 at α(t)) if and only if it is principal in S . 2 // 呜呜 好难QAQ -- 移居二次元(|R^2)的注意事项： 3. 如果你在从事random walk，往上下左右的 1. connectedness不保证pathwise connec- 的机率都是1/4，则你能回家的机率是1。 tedness。可能你跟你的幼驯染住很近， 4. 下面这个PDE是二次元上的波方程式 却永远没办法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不会出现 它的解不满足Huygens' principle，因此 chaos，在预测事情上比较方便。 讲话时会听到自己的回音，很不方便。 --

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