[試題] 105上 王金龍 幾何學 期末考
課程名稱︰幾何學
課程性質︰數學系選修 可抵數學系必修幾何學導論
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）︰106.1.13
考試時限（分鐘）：220 minutes
試題 :
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.
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正妹也不過就是一組物質波方程式的特解罷了
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