课程名称︰几何学
课程性质︰数学系大三必修
课程教师︰蔡宜洵
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/1/17
考试时限（分钟）：110
试题 :
Total points: 105
Notice: any theorems or facts put to use in the process of your proof or
computation must be stated clearly.
1 (20 pts in total). i) (15 pts) Given two (smooth) simple closed curves
C_1, C_2 on a sueface S of Gaussian curvature K < 0, such that they enclose a
region diffeomorphic to a cylinder. Suppose C_1 is a geodesic. Show that C_2
cannot be geodesic. ii) (5 pts) Let S be a compact surface without boundary
with K > 0 and C_1 be a simple closed geodesic which encloses a region R of S.
Suppose C_2, which belongs to R, is a smooth geodesic which divides R into two
connectes sets D_1, D_2 and is orthogonal to C_1 at the two points of
intersection. Show that the integral of the curvature
∫∫ K dσ = π, i=1,2.
D_i
2 (20 pts in total.) Let T be a torus of revolution parametrized by
x(u,v) = ((rcosu + a)cosv,(rcosu + a)sinv,rsinu). i) (15 pts) If a geodesic C
is tangent to the parallel u= π/2, then it is entirely contained in the
region given by -π/2 ≦ u ≦ π/2. ii) (5 pts) Is it true that C must
intersect the maximal parallel u = 0? Prove your answer.
3 (20 pts in total). Let S be a surface obtained by revolving the tractrix
z = g(x) with g'(x) =√1-x^2/x, g(1) = 0. i) (10 pts) Compute the Christoffel
symbols
— k
∣ ij, 1 ≦ i,j,k ≦ 2.
ii) (10 pts) Given a point p on S. Find a local parametrization in u,v such
that the first fundamental form around p is given by
2 2 2
ds = du + G(u,v)dv,
for an explicit G(u,v). Show the geometric meaning of these coordinates u,v.
2 2 2
4 (15 pts in total). Suppose ds = du + G(u,v)dv on S, with p = (0,0) and
√G(u,v)| = 1, √G (u,v)| = 0. Let a square Q_a be centered at p with
u=0 u u=0
vertices at (a,a), (-a,a), (-a,-a) and (a,-a). i) (10 pts) Show that around p,
2
√G = 1 - K(p)*u /2 modulo higher order terms in u,v. ii) (5 pts) Show that
K(p) =
8a-L(Q_a)
lim ___________ ,
a→0 3
2a
where L(Q_a) is the perimeter of Q_a.
5 (15 pts in total). Let C be a space curve α(t), t arc length parameter,
with k(t) =/= 0 for all t. Construct the tangent surface S as a ruled surface
with the rulings given by tangent lines to C. i) (8 pts) Given a
2
parametrization of S and find the first fundamental form ds of S (where it is
smooth). ii) (7 pts) Can it be (locally) isometric to a plane? Prove your
answer.
6 (15 pts in total). Let the unit sphere be parametrized by (sinψcosθ,
sinψsinθ,cosψ). Let C be a curve and w_p be a unit vector tangent to C at
some point p of C. For the parallel transport of w_p along C compute the angle
between the tangent ventor t_q and w_q at q in C in the following cases. i)
(10 pts) C is given by ψ = ψ_0, θ_1 ≦ θ ≦θ_2, with p at θ= θ_1 and
q at θ = θ_2. ii) (5 pts) C is given by (the equation) ψ = θ,
θ_1 ≦ θ ≦θ_2, with p at θ= θ_1 and q at θ = θ_2. (hint: you may use
Dw 1 dφ
[—] = ————{G v' - E u'} + — for ii).)
dt 2√EG u v dt