课程名称︰几何学 课程性质︰数学系选修 可抵数学系必修几何学导论 课程教师︰王金龙教授 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰106.1.13 考试时限（分钟）：220 minutes 试题 : 1. Given L(x, ξ), derive the Euler-Lagrange equations of t he action b . S[γ] = ∫ L(x, x) dt a i i for the end-points fixed γ. When x and p = ∂L / ∂x can be used as i coordinates for (x, ξ), prove its eqivalence with Hamilton's equations . . i x = ∂H / ∂p , p = - ∂H / ∂x , i i i where H(x, p) = Σ p ξ (x, p) - L(x, ξ(x, p)) is the energy. i i 2. Prove Cartan's homotopy formula L = ι d + d ι X X X . on forms, and show that for a Hamiltonian system y = ▽H(y) defined by a . general sympletic form Ω we have Ω = 0. 1 b .i .j 3. Show that the second variation for S[γ] = ---∫ Σ g x x dt is 2 a i,j ij b 2 G (ξ, η) = - ∫ 〈▽ ξ + R(ξ, T)T, η〉dt γ a T .i where γ is a geodesic and T = γ' = Σ x ∂. If R ≧ (n-1)K g for some i i ij ij constant K ＞ 0, show that a curve with length ＞ π/√K is not shortest. ij n 4. For the Hilbert-Einstein action S[g ] = ∫ R dσ = ∫ R√g d x, show that S S 1 δS 1 ----- ---------- = R - ---R g √g δg^{ij} ij 2 ij ij for the fast decayed g . Determine its energy-momentum tensor T . (If you ij k don't know how to do it, give its general definition and prove ▽ T = 0.) k i i 1 5. For A = Σ A (x) dx in Λ(g), where g is in M(n, R) is a matrix Lie i i algebra, define ▽ = ∂ + A and curvature F by i i i F(X,Y)ψ = ([▽ , ▽ ] - ▽ )ψ. Show X Y [X, Y] 2 (1) F is a tensor and F = dA + AΛA in Λ(g), i.e. F = ∂A - ∂A + [A , A ] ij i j j i i j (2) ▽ B = ∂ B + [A , B] for B(x) a g-valued function. i i i k k+1 (3) Bianchi's identity d F = 0, where d = D: Λ(g) → Λ (g) extends ▽. A A (4) For a non-degenerate Killing form 〈,〉 of g, any extremal A for ij S[A] = ∫ 〈F , F 〉dσ S ij with fast decay satisfies the Yang-Mills equation d*F = 0. A 6. Show anything "significant" in the last two chapters of the book or in my class which you have "well prepared" but not on the above. -- 正妹也不过就是一组物质波方程式的特解罢了 --

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