Integration on Manifolds via Volume Forms

1. From Euclidean Space to Manifolds: The Change of Variables Problem

Let's begin by revisiting the familiar integral in ${\mathbb{R}}^n$. The challenges we encounter when changing coordinates there will motivate our entire approach on manifolds.

The video below starts with a review of the Riemann integral and the change of variables formula. Pay close attention to the role of the Jacobian determinant.

The key point from the video is the change of variables formula for a scalar function $f$:
${\int }_{X}f(\mathbf{x})d^{n}x={\int }_{{\varphi }^{-1}(X)}f(\varphi (\mathbf{y}))|\det (J)|d^{n}y$
where $\mathbf{x}=\varphi (\mathbf{y})$ and $J$ is the Jacobian matrix $\frac{\partial x_i}{\partial y_j}$. The absolute value $|\det (J)|$ ensures that volume is always positive, regardless of whether the coordinate transformation preserves the "handedness" (orientation) of the coordinate system. We are integrating a measure.

However, as we saw in the last lesson, differential n-forms transform differently. If you have an n-form $\omega =f(\mathbf{x})d{x}^1\wedge \cdots \wedge d{x}^n$, its components transform under $\mathbf{x}=\varphi (\mathbf{y})$ as:
$d{x}^1\wedge \cdots \wedge d{x}^n=\det (J)d{y}^1\wedge \cdots \wedge d{y}^n$
Notice the absence of the absolute value. The transformation rule for an n-form naturally includes a sign that depends on the orientation of the transformation. This means that integrating an n-form is fundamentally tied to the concept of orientation.

This distinction is precisely what allows us to build a consistent theory of integration on manifolds.

2. Integration on Oriented Manifolds

To integrate an n-form on an n-dimensional manifold, we first need the manifold to be orientable.

An n-manifold $M$ is orientable if it can be covered by an atlas of charts such that on any overlap between two charts $(U_i,{\varphi }_i)$ and $(U_j,{\varphi }_j)$, the Jacobian of the transition map ${\varphi }_j\circ {\varphi }_i^{-1}$ has a positive determinant. An oriented manifold is an orientable manifold where we have chosen such an atlas. This choice defines a consistent "handedness" across the manifold.

With this, we can define the integral. The basic idea is to break the manifold into pieces that fit inside our coordinate charts, perform the standard Euclidean integral on each piece, and add up the results.

The following segments of the lecture by Frederic Schuller will guide us through this process.

We now have a way to define an integral over a region contained entirely within a single chart of an oriented atlas. But how do we integrate over the whole manifold, which may require many overlapping charts? Simply adding the integrals from each chart would lead to double-counting the regions where they overlap.

The elegant solution is a tool called a partition of unity. This is a collection of smooth functions $\{{\psi }_i\}$ associated with an atlas $\{U_i\}$ such that:

1. Each ${\psi }_i$ is non-zero only within its corresponding chart $U_i$.
2. At every point $p\in M$, the sum of all the functions is exactly 1: ${\sum }_i{\psi }_i(p)=1$.

A partition of unity allows us to smoothly "chop up" any function or form into pieces, each of which lives entirely within a single chart domain. We can then integrate each piece in its chart and sum the results without fear of double-counting.

The final segments of the video formalize this definition.

To summarize the formal definition: the integral of an n-form $\omega$ over an n-dimensional oriented manifold $M$ is given by
${\int }_M\omega ={\sum }_i{\int }_M{\psi }_i\omega ={\sum }_i{\int }_{{\varphi }_i(U_i)}(\text{pullback of }{\psi }_i\omega \text{ to }{\mathbb{R}}^n)$
where $\{{\psi }_i\}$ is a partition of unity subordinate to an oriented atlas $\{(U_i,{\varphi }_i)\}$.

3. The Metric Volume Form

So far, we can integrate n-forms. But in physics, we usually want to integrate scalar functions (like a Lagrangian density). To do this, we need to multiply the scalar function by a canonical n-form that represents the volume element of the space. This is the volume form.

For a general manifold, there is no unique choice of volume form. However, if the manifold is equipped with a metric $g_{\mu \nu }$, a natural volume form arises. This is the case for spacetime in General Relativity.

The metric volume form, often denoted $ϵ$, is defined in a coordinate system $x^{\mu }$ as:
$ϵ=\sqrt{|\det (g)|}d{x}^1\wedge \cdots \wedge d{x}^n$
where $g$ is the matrix of metric components $g_{\mu \nu }$.

• For a Riemannian manifold (like a spatial slice), $\det (g)>0$, so we use $\sqrt{\det (g)}$.
• For a Lorentzian manifold (spacetime), $\det (g)<0$, so we use $\sqrt{-\det (g)}$.

The miraculous property of this object is that it is coordinate-invariant. While neither $\sqrt{|\det (g)|}$ nor $d{x}^1\wedge \cdots \wedge d{x}^n$ are invariant on their own, their product is. The transformation of the determinant of the metric precisely cancels the Jacobian from the transformation of the one-form basis.

With the metric volume form, the integral of any scalar function $f$ over the manifold $M$ is now unambiguously defined:
${\int }_{M}fϵ=\int f(x)\sqrt{|\det (g)|}{d}^{n}x$
This is one of the most important practical formulas in General Relativity, appearing everywhere from the Einstein-Hilbert action to calculations of total mass.

Finally, the entire machinery of exterior derivatives and integration culminates in a single, powerful theorem: the Generalized Stokes' Theorem. It states that for any (n-1)-form $\omega$ on an n-dimensional manifold $M$ with boundary $\partial M$:
${\int }_{M}d\omega ={\int }_{\partial M}\omega$
This remarkable equation contains the fundamental theorem of calculus, Green's theorem, the classical Stokes' theorem, and the divergence theorem as special cases. It is the ultimate expression of the relationship between differentiation (d) and integration (∫) on manifolds.

Conclusion

This lesson marks the end of our first major module on the mathematical language of General Relativity. We have journeyed from the basic concept of a smooth manifold to the sophisticated machinery of tensor calculus, differentiation, and now, integration.

Key Takeaways:

• Integration on a manifold requires a notion of "volume," which is provided by a volume form: a non-vanishing, top-dimensional n-form.
• Integrating an n-form requires an oriented manifold. The definition relies on using a partition of unity to break the integral into pieces that can be computed within coordinate charts.
• When a manifold has a metric $g_{\mu \nu }$, it comes with a natural metric volume form, $ϵ=\sqrt{|\det (g)|}{d}^{n}x$.
• This allows us to define the integral of a scalar function $f$ in a coordinate-invariant way: ${\int }_{M}f:=\int f(x)\sqrt{|\det (g)|}{d}^{n}x$.
• The Generalized Stokes' Theorem, ${\int }_{M}d\omega ={\int }_{\partial M}\omega$, unifies all integral theorems of vector calculus into a single elegant statement.

You have now built the complete toolkit of modern differential geometry. In our next lesson, we will begin Module 2: Curvature and Dynamics. We will introduce the final piece of geometric structure needed for physics—the affine connection—which allows us to compare vectors at different points, define derivatives that account for the curvature of spacetime, and ultimately describe how particles move through it.