How to Construct a Smooth Pullback Function on a 2-Sphere Manifold

Hello! This is a beautiful problem that connects differential geometry, topology, and the theory of differential forms. Let's work through it together step by step!

1. What We're Solving:

We need to find a smooth function g on the 2-sphere S² such that when we pull back a 2-form α from ℝ³ via a map f: S² → ℝ³, the result equals g times the standard volume form ω₀ on the sphere.

2. The Approach:

This problem is fundamentally about understanding how differential forms behave under pullbacks and how they relate to volume forms. The key insight is that any 2-form on a 2-dimensional manifold (like S²) can be written as a function times the volume form. We need to find what that function is!

Think of it this way: we're asking "how does the pullback f*α compare to the natural volume element on the sphere?" The function g tells us the ratio between these two quantities at each point.

3. Step-by-Step Solution:

Step 1: Understand what we have

• f: S² → ℝ³ is our map from the sphere into 3-space
• α is a 2-form on ℝ³
• ω₀ is the standard volume form on S² (this gives us area measurements)
• f*α is the pullback of α to S²

Step 2: Recall the key theorem On any oriented 2-dimensional manifold, every 2-form can be written uniquely as a smooth function times the volume form. This is because the space of 2-forms at each point is 1-dimensional!

Step 3: Set up coordinates Let's use spherical coordinates (θ, φ) on S². The standard volume form is: ω₀ = sin φ dθ ∧ dφ

Step 4: Express the pullback The pullback f*α will also be a 2-form on S², so we can write: f*α = h(θ,φ) dθ ∧ dφ

for some function h.

Step 5: Find the relationship Since f*α = gω₀, we have: h(θ,φ) dθ ∧ dφ = g(θ,φ) sin φ dθ ∧ dφ

Step 6: Solve for g Comparing coefficients: g(θ,φ) = h(θ,φ)/sin φ

4. The Answer:

The function g at any point p ∈ S² is given by:

g(p) = (f*α)(p) / ω₀(p)

In coordinates: g(θ,φ) = [coefficient of dθ ∧ dφ in f*α] / sin φ

More conceptually: g measures the ratio between the "volume density" that α induces on the sphere (via the pullback) and the natural volume density of the sphere itself.

5. Memory Tip:

Remember this as the "volume ratio function" - g tells you how much the pullback form f*α stretches or shrinks the natural volume element of the sphere at each point. When g > 1, the pullback is "concentrating" more volume there; when g < 1, it's "diluting" it!

This problem beautifully illustrates how pullbacks allow us to compare geometric quantities on different spaces. You're essentially asking: "If I project the 2-form α onto the sphere via f, how does it compare to the sphere's natural area element?" Pretty elegant! 🌟