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