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

FINAL EXAM

JANUARY 21, 1996

Problem 1. (5 pts) Suppose that manifold X is a union of open subsets X = X_{1} ⋃ X_{2} and A_{i} is an atlas in X_{i}, i = 1, 2, which defines a differential structure, so that in Y = X_{1} ⋂ X_{2} structures agree. Define a differential structure in X, which agrees with the structures in X_{1} and in X_{2}.

Problem 2. (5 pts) Give explanations why S^{n} is orientable.

Problem 3. (5 pts) Suppose that n-dimensional differential manifold X has nform ω which does not vanish at x for any x ∈ X. Prove that X is orientable.

Problem 4. (5 pts) Assume that f: X → Y is a differentiable map, ω_{1} ∈ Ω^{p}(X), ω_{2} ∈ Ω^{p}(Y), η = ω_{1} ∧ f^{∗}(ω_{2}). Assuming that ω_{1} is closed and ω_{2} is exact can we claim that

(1) η is closed ?

(2) η is exact ?

(3) η = 0 ? If we can, give an explanation.

Problem 5. (5 pts) Consider curve γ: [0, 1] → ℝ^{3}, t ↦ (t, t^{2}, cos(πt)), and 1form ω = p_{z}^{∗}(z dz), where p_{z}: ℝ^{3} → ℝ is the projection to the axis Z. Determine ∫_{γ} ω.

Problem 6. (8 pts) Assume that U, V are smooth vector fields on a differentiable manifold X.

(1) Prove that there exist a vector field W on X such that for any function f ∈ ℂ^{∞}(X) we have W (f) = V (U(f)) − U(V (f)). (Hint: Show that the right hand side of the equality is a differentiation at any point).

(2) Assuming that dim X = 2 find coordinates of the vector field W is some coordinate system in terms of coordinates of the fields U and V.

Problem 7. (8 pts) Consider 1-form ω which is defined as ω = \frac{du}{v^{2}} in the chart with coordinates u = \frac{x}{z}, v = \frac{y}{z} in ℝp^{2}. Consider the curve γ: [0, 1] → ℝp^{2}, γ(t) = [1 : t : 0]. Extend by continuity the form ω outside the given chart and determine ∫_{γ} ω.