Middle East Technical University Spring 1996 Math 505 Differentiable Manifolds

TENSOR FIELDS, DIFFERENTIAL

FORMS AND STOCKS THEOREM

Problem List 3

Problem 1. (2 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 2. (1 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 3. (3 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.

Problem 4. (3 pts) Write down in cylidrical coordinates in ℝ^{3} (1) vector field V (x, y, z) = y \frac{d}{dx} − x\frac{d}{dy} + (x^{2} + y^{2} + z^{2})\frac{d}{dz},

(2) 1-form ω = y dx − x dy + (x^{2} + y^{2} + z^{2}) dz

(3) The Riemannian tensor of the standard metric in ℝ^{3}, which is given by the quadratic form g(v) = ∂x^{2} + ∂y^{2} + ∂z^{2}, v ∈ T_{p}ℝ^{3} ≅ ℝ^{3}.

Problem 5. (3 pts) Consider a vector field V and let T (ij) = \frac{∂V^{i}}{∂x^{j}}, where V = (V^{i}) in a coordinate system x = (x^{i}).

(1) Determine if T (ij) is a tensor of some type (i.e. the coordinates of T vary as coordinates of a tensor as we vary a coordinate system).

(2) Determine if S(i) = (V^{i})^{2} is a tensor field of some type.

(3) Determine if ∑_{i=1}^{n} T (i, i) is a well defined function (i.e. its values are independent of the choice of chart).

Problem 6. (2 pts) Determine the form ω′ ∈ Ω^{2}(ℝ^{3} − {O}), O = (0, 0, 0), which is the pull back of ω ∈ Ω^{2}(ℝp^{2}) with respect to projection p: ℝ^{3} − {O} → ℝp^{2}, p(x, y, z) = [x : y : z], if ω is defined in the chart (U_{z}, φ_{z}), u = \frac{x}{z}, v = \frac{y}{z}, as ω = ∂u ∧ ∂v. Determine dω′.

Problem 7. (4 pts) Let X be a smooth compact oriented manifold of dimension n with non-empty boundary ∂X. Prove that:

(1) there exists a differential form ω of degree n−1 on ∂X such that ∫_{∂X} ω ≠ 0;

(2) for any smooth map ρ: X → ∂X and any differential form ω of degree n−1 on ∂X one has dρ^{∗}ω = 0;

(3) if there exists a smooth retraction ρ: X → ∂X (i.e. a map, which is identity on the boundary), then ∫_{∂X} ω = 0 for any differential form ω of degree n−1 on ∂X.

How are these three statements related to the Borsuk theorem, which claims that there is no retractions X → ∂X ?

Problem 8. (3 pts) Let M, N be smooth manifolds of dimensions m, n respectively (m ⩾ n), and let f: M → N be a proper submersion (i.e., a proper surjective map whose differential is surjective at each point). Given an (m−n)-form ω on M, define f_{∗}ω: N → ℝ to be the function x ↦ ∫_{f^{−1}(x)} ω. Prove that if ω is closed then f_{∗}ω = const.