Integration on Manifolds - Core Definitions

Integration on manifolds extends the familiar notion of integration from calculus to curved spaces. Differential forms provide the natural objects to integrate, generalizing functions, vector fields, and volume elements.

DefinitionIntegration of n-Forms

Let $M$ be an oriented $n$ -manifold and $\omega$ a compactly supported $n$ -form on $M$ . The integral of $\omega$ over $M$ is defined by partitioning $M$ with charts $(U_\alpha, \varphi_\alpha)$ and a subordinate partition of unity $\{\rho_\alpha\}$ :

\int_M \omega = \sum_\alpha \int_{\varphi_\alpha(U_\alpha)} (\varphi_\alpha^{-1})^*(\rho_\alpha \omega)

where the right side uses standard Lebesgue integration on $\mathbb{R}^n$ .

This definition is independent of the choice of charts and partition of unity, provided the orientation is respected. The key is that $n$ -forms transform by the Jacobian determinant under coordinate changes.

ExampleIntegration on $\mathbb{R}^n$

On $\mathbb{R}^n$ with standard coordinates and orientation, integrating $\omega = f(x) dx^1 \wedge \cdots \wedge dx^n$ gives

\int_{\mathbb{R}^n} \omega = \int_{\mathbb{R}^n} f(x) dx^1 \cdots dx^n

recovering the usual Riemann or Lebesgue integral.

DefinitionIntegration over Submanifolds

Let $S \subset M$ be an oriented $k$ -dimensional submanifold and $\omega$ a $k$ -form on $M$ . The integral of $\omega$ over $S$ is

\int_S \omega = \int_S i^*\omega

where $i: S \hookrightarrow M$ is the inclusion map and $i^*\omega$ is the pullback, which is a $k$ -form on the $k$ -manifold $S$ .

Remark

The crucial fact is that only top-degree forms can be integrated on oriented manifolds. A $k$ -form integrates over a $k$ -dimensional oriented submanifold, giving a real number.

ExampleLine Integrals

A line integral $\int_\gamma \omega$ where $\gamma: [a,b] \to M$ is a curve and $\omega$ is a 1-form can be written as

\int_\gamma \omega = \int_a^b \gamma^*\omega = \int_a^b \omega_{\gamma(t)}(\gamma'(t)) dt

This generalizes work integrals $\int \mathbf{F} \cdot d\mathbf{r}$ from vector calculus.

DefinitionOrientation-Preserving Maps

A diffeomorphism $f: M \to N$ between oriented manifolds is orientation-preserving if for every chart $(U, \varphi)$ on $M$ and $(V, \psi)$ on $N$ in the oriented atlases, the Jacobian determinant of $\psi \circ f \circ \varphi^{-1}$ is positive.

TheoremChange of Variables

If $f: M \to N$ is an orientation-preserving diffeomorphism and $\omega$ is a compactly supported $n$ -form on $N$ , then

\int_M f^*\omega = \int_N \omega

If $f$ reverses orientation, then $\int_M f^*\omega = -\int_N \omega$ .

This is the manifold version of the change of variables formula from multivariable calculus, showing that integration respects the geometry encoded in differential forms.