課程名稱︰幾何學 課程性質︰數學系大三必修 課程教師︰崔茂培 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/01/09 考試時限（分鐘）：無限制 試題 : 3 (1) (10 pts) Let M be a compact surface contained in a closed ball in |R of radius R. Show that there exists at least one point p ∈ M where the Gauss -2 curvature and the absolute value of mean curvature are bounded below by R -1 and R respectively. (Hint: Consider the square of distance function from the surface to origin. Assume it achieves its maximum at p. We may assume that we can find a good parametrization σ(u, v) near p such that 2 (ⅰ) f(u, v) = ║σ(u, v)║ and f achieves its maximum at a point p. (ⅱ) 〈σ , σ 〉 = 〈σ , σ 〉 = 1 and 〈σ , σ 〉 = 0 at p. u u v v u v (2) (10 pts) Let k be a positive integer with k ≧ 2 and F(x, y, z) be a smooth k homogenous function of degree k, i.e., F(λx, λy, λz) = λ F(x, y, z) for all λ ∈ |R. Prove that away from the origin the induced metric on the conical surface M = {(x, y, z): F(x, y, z) = 0, (x, y, z) ≠ (0, 0, 0)} has Gaussian curvature equal to 0. (If you could not solve this problem, think 2 2 2 about the case F(x, y, z) = x - y - z first). 2 2 2 (3) (10 pts) Consider the complex curve in € defined by {(z, w) ∈ € : w - 3 z = α} for some α ∈ €. Under what condition(s) on α is this a two- dimensional manifold? (In real coordinates, z = z + iz and w = w + iw , 1 2 1 2 2 2 2 3 3 2 z , z , w , w ∈ |R. (a + ib) = a - b + i(2ab) and (a + ib) = a - 3ab 1 2 1 2 2 3 + i(3a b - b ).) (4) (8 pts) Show that if M is a compact n-manifold (without boundary) then M n does not embed in |R . 3 (5) (12 pts) Consider the map F: |R → GL(3, |R) by F(x, y, z) = z z ┌ e ze x ┐ │ z │ │ 0 e y │. Let S = {A ∈ GL(3, |R): A = F(x, y, z) for all (x, y, z) │ │ └ 0 0 1 ┘ 3 3 ∈ |R }. It is obvious that F: |R → S is a 1-1 and onto map. We define a multiplication on S by F(x, y, z)．F(u, v, w) = F(x, y, z) F(u, v, w) by matrix multiplication. (a) Show that S is a 3-dimensional Lie group. (b) Show that the left-invariant vector fields are spanned by z ∂ z ∂ z ∂ ∂ e = e F (──), e = ze F (──) + e F (──), e = F (──). 1 * ∂x 2 * ∂x * ∂y 3 * ∂z (c) Show that [e , e ] = 0, [e , e ] = -e , [e , e ] = -e - e . 1 2 1 3 1 2 3 1 2 2 (6) (8 pts) Could the sphere S be the underlying topological space of any Lie group? 3 4 2 2 2 2 (7) (15 pts) Let S = {(x, y, z, w) ∈ |R : x + y + z + w = 1}, and let ω 4 be the 1-form on |R given by ω = -ydx + xdy - wdz + zdw. 3 (a) Show that the restriction of the following vector fields to S are 3 independent tangent vector fields on S . ∂ ∂ ∂ ∂ -y ──+ x ──- w ──+ z──, ∂x ∂y ∂z ∂w ∂ ∂ ∂ ∂ -z ──+ w ──+ x ──- y──, ∂x ∂y ∂z ∂w ∂ ∂ ∂ ∂ -w ──- z ──+ y ──+ x──. ∂x ∂y ∂z ∂w (b) What are dω and ωΛdω? 3 (c) Prove that the restriction of the form ωΛdω to S is nowhere 0. ． 3 (d) Is ker(ω| ) integrable? here ker(ω| ) = ∪ {V ∈ T S : ω (V) S^3 S^3 p∈S^3 p p = 0}. (8) (10 pts) Suppose ω is a closed 2-form on a manifold M (i.e. dω = 0). For every point p ∈ M, let D (ω) = {v ∈ T M: ω (v, u) = 0 for all u ∈ T M}. p p p p ． Suppose that the dimension of D (ω) is the same for all p and D = ∪ D p p∈M p is a smooth distribution. Show that the distribution D is integrable. 3 (9) (7 pts) Show that a vector field X on |R has a flow (locally) that preserves volume (i.e. L Ω = 0 where the volume form is a 3-form Ω = dxΛ X dyΛdz) if and only if the divergence of X is everywhere 0, where the ∂ ∂ ∂ ∂f ∂g ∂h divergence of X = f ──+ g ──+ h ──is ──+ ──+──. ∂x ∂y ∂z ∂x ∂y ∂z (10)(10 pts) Let M be a smooth manifold. ∞ (a) Define a tri-linear map K: Ｘ*(M) ×Ｘ(M) ×Ｘ(M) → C (M) by K(ω, X, Y) = ω([X, Y]). Is K a smooth tensor field of type (1, 2)? ∞ (b) An almost complex structure J on M is a C (M)-linear map J: Ｘ(M) → 2 Ｘ(M) such that J(fX + gY) = fJ(X) + gJ(Y) and J (X) = -X for X, Y ∈ ∞ Ｘ(M) and f, g ∈ C (M). The Nijenhuis tensor N is defined by N(X, Y) = [X, Y] + J[JX, Y] + J[X, JY] - [JX, JY] for X, Y ∈ Ｘ(M). Define a tri-linear map ∞ N : Ｘ*(M) ×Ｘ(M) ×Ｘ(M) → C (M) J by N (ω, X, Y) = ω(N(X, Y)). J Prove that N is a smooth tensor field of type (1, 2). J -- 移居二次元(|R^2)的注意事項： 3. 如果你在從事random walk，往上下左右的 1. connectedness不保證pathwise connec- 的機率都是1/4，則你能回家的機率是1。 tedness。可能你跟你的幼馴染住很近， 4. 下面這個PDE是二次元上的波方程式 卻永遠沒辦法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不會出現 它的解不滿足Huygens' principle，因此 chaos，在預測事情上比較方便。 講話時會聽到自己的回音，很不方便。 --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.212.7 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1452414779.A.E20.html