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）︰2015/11/13
考試時限（分鐘）：165
試題 :
Total = 110 points. Enjoy the exam on black Friday!!!
2
(1) (20 pts) A regular spherical curve γ(s): I → S (1) parametrized by arc-
length has an alternate set of equations that describe its properties. Note
that γ(s) is also normal to the sphere, then the signed normal can be
defined as the vector S = γ ×T. Then {T, S, γ} becomes an orthonormal
frames along γ(s).
(a) (6 pts) Show that
dT
─ = κ S - γ
ds g
dS
─ = -κ T
ds g
dγ
──= T
ds
dT
where the geodesic curvature κ is defined as κ = ─．S.
g g ds
(b) (10 pts) Show that
dκ
2 2 1 g
κ = 1 + κ , τ = ──────
g 2 ds
1 + κ
g
1 1
N = ─(κ S - γ), B = ─(κ γ + S).
κ g κ g
(c) (4 pts) Show that γ is planar if and only if the curvature is constant.
(2) (15 pts) For a regular space curve γ(s) parametrized by arc-length, we say
dX
that a normal field X is parallel along γ if X．T = 0 and ─ is parallel to
ds
T.
(a) (3 pts) Show that for a fixed s and X(s ) ⊥ T(s ) there is a unique
0 0 0
parallel field X that is X(s ) at s .
0 0
(b) (6 pts) A Bishop frame consists of an orthonormal frame T, N , N along
1 2
the curve so that N , N are both parallel along γ. For such a frame
1 2
show that
dT
─ = κ N + κ N
ds 1 1 2 2
dN
1
──= -κ T
ds 1
dN
2
──= -κ T
ds 2
Note that such frames always exist, even when the space curve doesn't
have positive curvature everywhere.
2 2 2
(c) (3 pts) Show further that for such a frame κ = κ + κ .
1 2
(d) (3 pts) Show that if γ has positive curvature so that N is well-
dφ
defined, then N = cosφ N + sinφ N , B = -sinφ N + cosφ N and ──
1 2 1 2 ds
= τ where φ(s) is defined by κ = κcosφ and κ = κsinφ.
1 2
(3) (15 pts) Recall that the first fundamental of a regular surface σ can be
╭ σ ．σ σ ．σ ╮
│ u u u v │
identified with the matrix Ⅰ = │ │, the second
│ σ ．σ σ ．σ │
╰ u v v v ╯
fundamental can be identified with the matrix Ⅱ =
╭ -σ ．N -σ ．N ╮
│ u u u v │
│ │. We can define the third fundamental as the first
│ -σ ．N -σ ．N │
╰ v u v v ╯
╭ N ．N N ．N ╮
│ u u u v │
fundamental form of the Gauss map N, i.e., Ⅲ = │ │. Prove
│ N ．N N ．N │
╰ u v v v ╯
that Ⅲ - 2HⅡ + KⅠ = 0.
(4) (10 pts) If the first fundamental form of the surface is I =
2 2 -1 2 2
(1 + u + v ) (du + dv ), compute the Gaussian curvature of the surface.
(5) (20 pts) Let C be a circle of radius 10 that contains the points (0, 0, 8)
and (0, 0, -8), and let A be the (open) minor arc of C between these points.
Let S be the surface obtained by rotating the arc A around the z-axis,
oriented so that normal vectors point outwards.
http://i.imgur.com/F11kUGK.png
Note that the surface S does not include the cusp point (0, 0, 8) and (0, 0,
-8).
(a) Find the principal curvatures of S at the point (4, 0, 0).
(b) Find the image of S under the Gauss map. Express your answer as one or
more inequalities defining a region on the unit sphere.
(c) Use your answer to part (b) to evaluate ∫ K dA, where K is the Gaussian
S
curvature of S. (Use geometry to find the answer.)
3
(6) (10 pts) Suppose that M is a compact orientable surface in |R and K > 0
everywhere on M.
(a) What can you say about the topology of M and why?
(b) Show that the Gauss map is a one-to-one and onto map.
3
(7) (10 pts) If M is a compact orientable surface in |R and has constant
Gaussian curvature, then it is a round sphere.
(8) (10 pts) Suppose that a curve γ lies in two surfaces S and S that make a
1 2
constant angle along γ (i.e. their tangent plane make a constant angle).
Show that α is principal in S (i.e. α'(t) is a principal direction for S
1 1
at α(t)) if and only if it is principal in S .
2
// 嗚嗚 好難QAQ