課程名稱︰幾何分析概論
課程性質︰數學系選修
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）︰2016/04/22
考試時限（分鐘）：100
試題 :
There is a 20% discount for those problems that you turn in after the in-
class exam.
This is an open book exam. You can use the textbooks, notes, and computers
during the exam. But you can not use google search during the exam. Please write
down the details and explain your argument carefully.
2n-1 n
Let n ≧ 2 and f: S → S be a smooth map between oriented spheres. Let ω
n
be a smooth n-form on S with ∫ ω = 1.
S^n
n n
(a) (15 pts) Show that [ω] ∈ H (S , |R) ≠ 0.
2n-1
(b) (15 pts) Show that f*ω = dγ for some (n-1)-form γ on S .
(c) (20 pts) With notation as above, define H(f) = ∫ γ Λ dγ. Show
S^{2n-1}
that H(f) is independent of the choices of γ and ω.
(d) (15 pts) Show that H(f) = H(g) if f is homotopic to g.
(e) (20 pts) Show that if n is odd, then H(f) is zero for any f.
3 2
(f) (20 pts) If F: S → S is the Hopf map defined by
2 2 2 2
F(X , X , X , X ) = (2(X X + X X ), 2(X X - X X ), (X + X ) - (X + X )).
1 2 3 4 1 2 3 4 1 4 2 3 1 3 2 4
4
As a submanifold of |R , the 3-sphere is
3 2 2 2 2
S = {(X , X , X , X ): X + X + X + X = 1},
1 2 3 4 1 2 3 4
3
and the 2-sphere is a submanifold of |R ,
2 2 2 2
S = {(x , x , x ): x + x + x = 1}.
1 2 3 1 2 3
Find H(F).
(Hint: You may want to choose the 2-form to be
1 1 2 3 2 1 3 3 1 2
ω = ──(x dx Λdx - x dx Λdx + x dx Λdx ).
4π
In general,
︿
n+1 i-1 i 1 i n+1
Ω = Σ (-1) x dx Λ...Λdx Λ...Λdx ,
i=1
i i
where the caret ^ over dx indicates that dx is to be omitted. Recall that
n n+1 n+1
Ω is an orientation form on S with ∫ Ω = (n+1) Vol(D ), where D =
S^n
n+1 4 2 3
{x ∈ |R : ║x║ < 1}. Vol(D ) = π /2 and Vol(D ) = 4π/3.)