Middle East Technical University Spring 1996 Math 505 Differentiable Manifolds

MIDTERM EXAMINATION

APRIL 20

TIME: 3 HOURS

Problem 1. (2 pts) Prove that the quotient map S^{2} → ℝp^{2} is differentiable.

Problem 2. (3 pts) Let D_{0} be the standard smooth structure on ℝ. Construct another smooth structure D on the same topological space so that the identity map id: (ℝ, D_{0}) → (ℝ, D) is not differentiable. Construct a diffeomorphism (ℝ, D_{0}) → (ℝ, D).

Problem 3. (3 pts) Consider function f: U_{0} → ℂ defined in the coordinate system (U_{0}, ϕ_{0}) on the Riemannian sphere ℂp^{1}, U_{0} = {[z_{0} : z_{1}] ∈ ℂp^{1} | z_{0} ≠ 0}, z = ϕ_{0}([z_{0} :z_{1}]) = \frac{z_{1}}{z_{0}} , as f(z) = z^{3} + z − 2.

(1) Write down f in another chart on ℂp^{1}.

(2) Determine if f can be extended to a smooth function defined on the whole ℂp^{1}.

(3) Determine if f can be extended to a smooth map f̄: ℂp^{1} → ℂp^{1}.

Problem 4. (3 pts) Consider the map f: ℝ^{2} → ℝp^{2}, (x,y) ↦ [x^{2} −y^{2} : x+y : x^{2} + 1]. Write down f using some chart in ℝp^{2} and the polar coordinate system in ℝ^{2}.

Problem 5. (4 pts) Consider point p ∈ ℝp^{2} with coordinates [1:3:1], covector α ∈ T_{p}^{∗}(ℝp^{2}) with coordinates (−1, 1) in the chart φ_{x}: U_{x} → ℝ^{2}, φ_{x}[x: y: z] = (\frac{y}{x}, \frac{z}{x}), where U_{x} = {[x:y:z] | x ≠ 0}, and vector v tangent to the curve γ: [−1, 1] → ℝp^{2}, γ(t) = [t+1 : 2t^{2}+3 : e^{t}] at p (t = 0).

(1) Determine the coordinates of p in the charts φ_{x}, φ_{y} and φ_{z}.

(2) Determine the coordinates of α in the chart φ_{y}.

(3) Determine the coordinates of v in the chart φ_{z} and evaluate α(v).

Problem 6. (3 pts) Given covectors α_{1}, …, α_{k} ∈ V^{∗}, where V is n-dimensional vector space, n ⩾ k, prove that a_{1} ∧ … ∧ α_{k} = 0 if and only if α_{1}, …, α_{k} are linear dependent.

Problem 7. (3 pts) Consider a tensor field T of type (p, q) on ℝ, which has (its unique) coordinate T (x) = T_{1,...,1}^{1,...,1}(x) = x^{2} in the identity chart id: ℝ → ℝ. Determine the coordinate of T in the chart ϕ: ℝ → ℝ, ϕ(x) = 2x.

Problem 8. (5 pts) Let F_{t}, G_{s}: M → M, t, s ∈ ℝ, be 1-parameter group of diffeomorphisms defined by the vector fields X and Y on a manifold M. Assuming that F_{t} ∘ G_{s} = G_{s} ∘ F_{t}, ∀t, s ∈ ℝ, prove that [X, Y] = 0 at any point.