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）︰105.12.2 考試時限（分鐘）：220 minutes 試題 : 1. [Lie derivative] (20 pts) (a) Define L_ξ T of a tensor field T along a vetor field ξ and compute L_ξ T ^{i_1,...,i_p}_{j_1,...,j_q}. (b) Show that L_ξ η = [ξ, η] for vector field η and L_ξ d = d L_ξ on differential forms. 2. [Cartan d and Hodge * in n dimensional Riemannian space] (20 pts) (a) Define *: Λ^p → Λ^{n-p} by 1 i_1 ... i_p (*T) = --- √g ε T . i_p+1 ... i_n n! i_1 ... i_n Show that *^2 = (-1)^p(n-p) and T^(*S) = {T, S}dσ. p (b) On Ω , forms supported in a bounded region U, with 〈ω_1, ω_2〉 U defined as ∫ ω_1 ^ *ω_2, show that the adjoint of d is given by δ, U defined as (-1)^(np-n+1) *d* and δ^2 = 0. (c) Let △ = dδ + δd. Show that △ is self-adjoint and it commutes with d, δ and *. (d) △ω = 0 if and only if dω = 0 and δω = 0. If furthermore ω = dη, then ω = 0. 3. [Invariant metric on classical Lie groups] (20 pts) (a) For a matrix group G with X in g, show that R_X defined by R_X(A) = -XA for A in G is right invariant. Also [R_X, R_Y] = R_[X, Y] and [L_X, R_Y] = 0. (b) Show that a left invariant metric 〈,〉 defined by a Killing form is bi-invariant and Ω(L_X, L_Y, L_Z) defined by 〈[L_X, L_Y], L_Z〉 is a 3-form with dΩ = 0. LC (c) With 〈,〉 in (b), determine ▽ and all geodesics through e in G. 4. [Geodesic normal coordinates] (20 pts) n (a) Show that exp : R (isomorphic to T ) → U defined by geodesics p p ξ → γ (1) is invertibla near p, and the connections and ∂g all vanish ξ k ij at p. (b) For a surface U with polar coordinate (ρ, θ), induced from T , show that p F = 〈∂ , ∂ 〉 = 0. Also G(0, θ) = 1, √G (0, θ) = 0 and ρ θ ρ K = -√G / √G . ρρ 5. [Pseudo-Riemannian space with Levi-Civita connection] (20 pts) i i (a) Prove the two Bianchi identities R = 0 and R = 0. [jkl] j[kl;m] i 1 (b) Show that ▽ R = ---∂R. If R = λg , when can we deduce that i m 2 m ij ij λ is a constant? (c) For n = 3, show that R is determined by R . ijkl ij 6. [Gauss-Bonnet theorem on a surface] (20 pts) (a) Let α be a piecewise smooth closed curve bounding a region Ω in a surface. Prove that ∫ K dA = θ , where θ is the holonomy angle along Ω α α the curve. (b) Prove the local Gauss-Bonnet Theorem 2π = Σ α + ∫ k dl + ∫ K dA. outer angles j ∂Ω g Ω -- 正妹也不過就是一組物質波方程式的特解罷了 --

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