Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 8 April 10, 2014

Problem 1: (Spherical coordinate system) Let Φ : R^{3} ⟶ R^{3} be defined by (r, ϑ, φ) ↦ (x, y, z) = (r sin ϑ cos φ, r sin ϑ sin φ, r cos ϑ). In an open set V ⊂ R^{3} let

ω_{1} = f_{1} dx + f_{2} dy + f_{3} dz
ω_{2} = F_{1} dy ∧ dz + F_{2} dz ∧ dx + F_{3} dx ∧ dy
ω_{3} = Σ dx ∧ dy ∧ dz.

Let U = Φ^{−1}(V) and

Φ^{∗}ω_{1} = g_{1} dr + g_{2} dϑ + g_{3} dφ
Φ^{∗}ω_{2} = G_{1} dϑ ∧ dφ + G_{2} dφ ∧ dr + G_{3} dr ∧ dϑ
Φ^{∗}ω_{3} = Ξ dr ∧ dϑ ∧ dφ.

Compute the functions g_{i}, G_{i}, Ξ : U ⟶ R.

Problem 2:

(a) Consider on R^{3} the 2-form ω = 2xz dy ∧ dz + dz ∧ dx − (z^{2} + e^{x}) dx ∧ dy.

Show that ω is closed and determine a 1-form η with dη = ω.

(b) Consider on R^{2} \ {0} the 1-form

ω = \frac{x dy − y dx}{x^{2} + y^{2}}

Show that ω is closed but not exact. (Hint: Solve ω = dη locally and show that η cannot be extended to a globally defined 1-form.)

Problem 3:

Two differentiable maps f, g : M ⟶ N between differentiable manifolds are said to be differentiable homotopic if there exists an open subset V ⊂ R × M containing [0, 1] × M ⊂ V and a differentiable map F : V ⟶ N such that f(x) = F (0, x) and g(x) = F (1, x) for x ∈ M.

Prove that diffentiable homotopic maps f and g induce the same map f^{∗} = g^{∗} : H_{DR}^{∗}(N) ⟶ H_{DR}^{∗}(M) between the de Rham cohomology groups.

(Hint: Construct maps h_{r} : E^{r}(N) ⟶ E^{r−1}(M) for all r such

h_{r+1} ∘ d + d ∘ h_{r} = f^{∗} − g^{∗}.)

Problem 4: (Symplectic structure on cotangent bundle)

Let M be a n-dimensional differentiable manifold and π : T^{∗}M ⟶ M its cotangent bundle. Let θ ∈ E(T^{∗}M) be the 1-form on T^{∗}M which sends a vector in T (T^{∗}M) to ev(σ, T π) where σ : T (T^{∗}M) ⟶ T^{∗}M is the natural projection, T π : T (T^{∗}M) ⟶ T M is the differential of π and ev : T_{p}^{∗}M × T_{p}M ⟶ R is the natural evaluation map of a cotangent vector at a tangent vector for a point p ∈ M. Note that this is well-defined since for u ∈ T (T^{∗}M) the elements σ(u) and T π(u) are in the cotangent resp. tangent space of the point π(σ(u)) ∈ M.

Consider the closed 2-form ω = dθ ∈ E^{2}(T^{∗}M).

(a) Show that in local coordinates x_{1}, …, x_{n} around p ∈ M one has

ω = x_{1} ∧ y_{1} + ⋯ + x_{n} ∧ y_{n}

where y_{1} = dx_{1}, …, y_{n} = dx_{n} and the x_{i}, y_{i} are the used local coordinates for T^{∗}M.

(b) Show that ω^{n} := \underbrace{ω ∧ ω ∧ … ∧ ω}_{n−times} is a nowwhere vanishing 2n-form on

T^{∗}M.