Questions: Integration on Manifolds

Question 1

To define ∫_M ω for an n-form ω on an oriented n-manifold M, you need a partition of unity {ρα} and write ∫_M ω = Σα ∫_M ρα ω. Each term ∫_M ρα ω is computed by...

• A. Evaluating ρα ω at a single point and multiplying by the volume of M
• B. Pulling back ρα ω to ℝⁿ via the coordinate chart φα and computing a standard Riemann/Lebesgue integral
• C. Using the metric to convert ω to a function and integrating that function
• D. Approximating M by a simplicial complex and summing over simplices

Answer: (B)

Since supp(ρα ω) ⊂ Uα, we pull back to the coordinate chart: ∫_M ρα ω = ∫_{φα(Uα)} (φα⁻¹)*(ρα ω), which is an integral of an n-form on an open subset of ℝⁿ — a standard multivariable integral. In coordinates, if ρα ω = f dx¹ ∧ ... ∧ dxⁿ, this becomes ∫ f dx¹...dxⁿ. No metric is needed — the form itself provides the 'volume element.' The partition of unity ensures independence of the chosen cover and partition.

Question 2

Reversing the orientation of M multiplies ∫_M ω by -1.

• T. True
• F. False

Answer: True

An orientation determines which coordinate charts are 'positively oriented' — those with positive Jacobian determinant transitions. Reversing orientation flips the sign of the Jacobian determinant in the change-of-variables formula, which flips the sign of the integral. Equivalently, if -M denotes M with the opposite orientation, then ∫_{-M} ω = -∫_M ω. This is why oriented manifolds are essential: without orientation, the integral of a form is not well-defined (it could be either sign).

Question 3

Integration on manifolds does not require a Riemannian metric. What, then, determines the 'volume element' for integration?

Model answer: The differential form itself serves as the volume element. An n-form ω on an n-manifold assigns to each infinitesimal parallelepiped (spanned by n tangent vectors) a signed volume. No metric is needed to define this — the form evaluates on tangent vectors directly. What is needed is an orientation (to fix the sign) and the form to integrate. A Riemannian metric becomes necessary only when you want a canonical volume form (the Riemannian volume form dVg = √det(gij) dx¹∧...∧dxⁿ) or when you want to integrate functions rather than forms.

This is a key conceptual point. In vector calculus, the volume element dV = dx dy dz seems to require knowing what 'volume' means, which suggests a metric. But differential forms carry their own notion of volume. The metric enters only when you want to measure lengths, angles, or integrate scalar functions (in which case you multiply the function by the metric volume form). The metric-free nature of form integration is what makes Stokes' theorem work in full generality.