Middle East Technical University Fall 1995 Math 471 Elementary Topology and Geometry

DIFFERENTIABLE MANIFOLDS

Problem List 2

Problem 1. (1 pt) For the map f: ℝp^{2} → ℝp^{1}, [x:y:z] ↦ [x^{2}+y^{2}+z^{2} :xy+yz+zx], and the tangent vector v ∈ T_{p}(ℝp^{2}) which has coordinates (1, −1) in chart φ_{x}, p = [1 : 2 : 3], determine coordinates of d_{p}f(v) in the chart φ_{a}, [a : b] ↦ \frac{b}{a}.

Problem 2. (1 pt) Vector v ∈ T_{p}(ℂp^{1}), p = [2 : i] has coordinate (1 + i) in the chart [z : w] ↦ \frac{z}{w}. Determine D_{v}(f) for the function f which is defined by f(u) = u^{2} + u − 1 in the opposite chart, [z : w] ↦ \frac{w}{z}.

Problem 3. (3 pts) Let V be the vector field on P = {(x, y) | y = x^{2}} formed by the unit length tangent vectors directed to the right, p_{X} , p_{Y} : P → ℝ denote projections to OX and OY.

(1) Write down V in the coordinate system p_{X}.

(2) Determine the function V (p2_{Y}).

(3) Determine ω(V), where ω = p∗_{Y} dy.

Problem 4. (3 pts) Assume that f: X → Y is a homeomorphism and X is a differential manifold.

(1) Define a differential manifold structure on Y.

(2) Determine if f is diffeomorphism with respect to these structures.

Problem 5. (3 pts) Consider the graph Γ of the map ℝ → ℝ, x ↦ |x|. (1) Define an atlas which gives a differential structure on Γ.

(2) Prove that the inclusion of Γ into ℝ^{2} is not embedding.

(3) Construct another smooth structure on Γ such that the identity map Γ → Γ is not smooth.

Problem 6. (5 pts) Consider the map, f: ℝp^{2} → ℝ, f([x: y : z]) ↦ \frac{xy+z^{2}}{x^{2}+y^{2}+z^{2}} , and vector field V on ℝp^{2}, defined as V (u, v) = 2u\frac{∂}{∂u} + (uv − v)\frac{∂}{∂v} in the chart φ_{x}.

(1) Determine the maximal domain in ℝp^{2} to which V can be smoothly extended. In particular, determine if V can be extended to the whole ℝp^{2}.

(2) Determine V (f).

(3) Determine f^{∗}(ω) in the chart φ_{y} if ω(t) = (2t + 1) dt is a 1-form in ℝ.

(4) Determine ω(V).