MTH4111/5111-MTH5111 /4111-Differential geometry代写
时间:2023-07-30
MTH4111/5111 ASSIGNMENT 1
Important
(a) This assignment is due at 11:55 pm on the Sunday of Week 3. All assignments are
to be submitted electronically via Moodle and must be submitted as a single PDF file.
If your assignment is handwritten, then please ensure that the scan of your assignment is
clearly legible; illegible or difficult to read assignments will not be graded.
(b) MTH4111 students: Only the first the first 2 questions will be graded for students
enrolled in MTH4111. The third problem is optional and will not count for marks.
(c) Late penalties apply: 10% per day until the solutions are released after which the as-
signment is worth zero and will only be marked for feedback purposes.
(d) You can talk with the other students and myself about the problems, but you must write
up and hand in your own work.
(e) I expect that you will put enough thought and effort into the presentation of your solutions
so that they are neat, clear, and concise. Poorly presented assignments will be penalized.
Problem 1. Let RPn be the quotient space of1 Rn+1× by the following equivalence relation: for
x, y ∈ Rn+1× , x ≈ y ⇔ x = λy for some λ ̸= 0.
(i) Let π : Rn+1× → RPn be the canonical projection map, that is
π(x0, x1, . . . , xn) := [x0, x1, . . . , xn],
where [x0, x1, . . . , xn] denotes the equivalence class of the point (x0, x1, . . . , xn) ∈ Rn+1× .
For i = 0, . . . , n, let Ui = {[x0, x1, . . . , xn] ∈ RPn |xi ̸= 0} and define ϕi : Ui → Rn by
ϕi([x
0, x1, . . . , xn]) =
(
x0
xi
,
x1
xi
, . . . ,
x̂i
xi
, . . . ,
xn
xi
)
,
where x̂
i
xi
means it is omitted. Show that ϕi is a well-defined bijection. [3 marks]
(ii) For i, j ∈ {0, . . . , n} i ̸= j, compute the transition maps
ϕj ◦ ϕ−1i : ϕi(Ui ∩ Uj) −→ ϕj(Ui ∩ Uj)
and show that they are smooth (in the usual sense as maps between open sets of Rn).
[3 marks]
(iii) Show that A = ∪ni=0{(Ui, ϕi)} is an atlas for RPn, and hence, that RPn is an n-
dimensional, smooth manifold, known as real projective n-space .
[3 marks]
(iv) Show that the projection π : Rn+1× → RPn is smooth. [3 marks]
Problem 2. Let F : Rn+1× −→ Rm+1× be a smooth function, and suppose that for some k ∈ Z,
F (λx) = λkF (x) for all x ∈ Rn+1× and λ ∈ R×. Show that the map f : RPn× −→ RPm defined
by f([x]) := [F (x)] is well-defined and smooth. [10 marks]
1Here, we are using the notation V× := V \ {0}, where V is a vector space.
1
2 MTH4111/5111 ASSIGNMENT 1
Problem 3. Suppose that S ⊂M is a closed submanifold of M and suppose that f ∈ C∞(S).
Then show that there exists a f˜ ∈ C∞(M) such that f˜ |S = f . Is it still true that the smooth
function f on S can be extended to M if S is not closed? [10 marks]