Integration on Manifolds and Lie Groups

This note fills in some details for §5 in Chapter I of Bröcker–tom Dieck.

First, we review some basic material on integration on manifolds.

Integration on a smooth manifold Let M be an oriented smooth n-manifold, and denote by C_{c}^{n}(M) the vector space of continuous n-forms on M with compact support. Our object here is to define the integral ∫_{M}: C_{c}^{n}(M) → ℝ.

Take a smooth atlas on M consisting of orientation-preserving diffeomorphisms h_{λ}: U_{λ} ≅ U′_{λ} ⊂ ℝ^{n}, where the open sets U_{λ} (for λ ∈ Λ) cover M. Take a partition of unity, consisting of continuous functions ϕ_{λ}: M → ℝ with support supp(ϕ_{λ}) ⊂ U_{λ}, so that ∑_{λ} ϕ_{λ} = 1. (We do not need ϕ_{λ} to be smooth for current purposes.)

Given ω ∈ C_{c}^{n}(M), we decompose it as the sum ω = ∑_{λ} ϕ_{λ}ω; since every point of M has a neighborhood on which only finitely many of the ϕ_{λ} are nonzero and supp(ω) is covered by finitely many such neighborhoods, this sum is effectively finite: all except finitely many terms are identically zero. We handle each term separately.

We pull back the n-form ϕ_{λ}ω by h_{λ}^{−1} to the open set U′_{λ} in ℝ^{n}. The canonical orientation of ℝ^{n} orients the standard basis (e_{1}, e_{2}, …, e_{n}) positively. We write the coordinates on ℝ^{n} as x = (x_{1}, x_{2}, …, x_{n}) (and recall that the coordinate x_{i} is really a function, the projection ℝ^{n} → ℝ to the i-th factor, so that dx_{i} is a 1-form). The result is

(h_{λ}^{−1})^{∗}(ϕ_{λ}ω) = a_{λ} dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n},

where the function a_{λ} is given explicitly as

a_{λ}(x) = (a_{λ} dx_{1} ∧ … ∧ dx_{n})_{x}(e_{1}, …, e_{n}) = ϕ_{λ}(p) ω_{p}(T (h_{λ}^{−1})e_{1}, …, T (h_{λ}^{−1})e_{n}) (1)

and p = h_{λ}^{−1}(x).

Definition 2 We define the integral in terms of multiple integrals on ℝ^{n}, as

∫_{M} ω = ∑_{λ} ∫ ⋯∫_{U′_{λ}}a_{λ}(x) dx_{1} dx_{2} … dx_{n}.

Remark If ω_{p}(v_{1}, v_{2}, …, v_{n}) ≥ 0 for one (and hence any) positively oriented basis of the tangent space T_{p}(M), equation (1) shows that a_{λ}(h_{λ}(p)) ≥ 0. If the condition holds for all p ∈ M, we deduce that ∫_{M} ω > 0 unless ω is everywhere zero.

Remark If we reverse the orientation of M, the local chart h_{λ} no longer preserves orientation. We may replace it by r ^{∘} h_{λ}, where r is the reflection of ℝ^{n} that reverses the first coordinate x_{1} only. Since r^{∗}dx_{1} = dr^{∗}x_{1} = −dx_{1}, we find

(h_{λ}^{−1}^{∘}r)^{∗}(ϕ_{λ}ω) = r^{∗}(a_{λ} dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n}) = −(r^{∗}a_{λ}) dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n}.

However,

∫ ⋯∫ a_{λ}(r(x)) dx_{1} dx_{2} … dx_{n} = ∫ ⋯∫ a_{λ}(x) dx_{1} dx_{2} … dx_{n}.

The net effect is to change the sign of ∫_{M} ω.

Theorem 3 Given an oriented smooth n-manifold M, Definition 2 yields a canonical ℝ-linear function

∫_{M}: C_{c}^{n}(M) − ; − ; → ℝ.

Proof ℝ-linearity is obvious. We have to show that the integral is independent of the choices of atlas and partition of unity.

Take another atlas, consisting of orientation-preserving diffeomorphisms k_{μ}: V_{μ} ≅ V′_{μ} ⊂ ℝ^{n}, with associated partition of unity consisting of functions ψ_{μ}. We decompose ω = ∑_{λ} ∑_{μ} ϕ_{λ}ψ_{μ}ω, so that ϕ_{λ}ψ_{μ}ω has compact support in U_{λ} ∩ V_{μ}. We pull back each term,

(h_{λ}^{−1})^{∗}(ϕ_{λ}ψ_{μ}ω) = a_{λ,μ} dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n}.

It is clear that a_{λ} = ∑_{μ} a_{λ,μ}. Similarly,

(k_{μ}^{−1})^{∗}(ϕ_{λ}ψ_{μ}ω) = b_{μ,λ} dy_{1} ∧ dy_{2} ∧ … ∧ dy_{n},

where we use coordinates (y_{1}, y_{2}, …, y_{n}) on this copy of ℝ^{n} to avoid confusion. It will be sufficient to prove that

∫ ⋯∫ b_{μ,λ}(y) dy_{1} dy_{2} … dy_{n} = ∫ ⋯∫ a_{λ,μ}(x) dx_{1} dx_{2} … dx_{n}.

We have the diffeomorphism θ = k_{μ} ^{∘} h_{λ}^{−1}: h_{λ}(U_{λ} ∩V_{μ}) ≅ k_{μ}(U_{λ} ∩V_{μ}) between open sets in ℝ^{n}, hence

θ^{∗}(b_{μ,λ} dy_{1} ∧ dy_{2} ∧ … ∧ dy_{n}) = a_{λ,μ} dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n},  (4)

as both sides are pullbacks from ϕ_{λ}ψ_{μ}ω on U_{λ} ∩ V_{μ} ⊂ M. The left side is

(θ^{∗}b_{μ,λ})(θ^{∗}dy_{1}) ∧ (θ^{∗}dy_{2}) ∧ … ∧ (θ^{∗}dy_{n}).

Here, (θ^{∗}b_{μ,λ})(x) = b_{μ,λ}(θ(x)). In coordinates, θ(x) = (θ_{1}(x), …, θ_{n}(x)) ∈ ℝ^{n}, where θ_{j} = y_{j} ^{∘} θ. Then we can write

θ^{∗}dy_{j} = dθ^{∗}y_{j} = d(y_{j} ^{∘} θ) = dθ_{j} = ∑_{i=1}^{n} \frac{∂θ_{j}}{∂x_{i}} dx_{i}.

Properties of ∧ show that (θ^{∗}dy_{1}) ∧ (θ^{∗}dy_{2}) ∧ … ∧ (θ^{∗}dy_{n}) expands to

J dx_{1} ∧ dx_{2} ∧ … ∧ dx_{n},

where J(x) denotes the Jacobian determinant of θ at x. Thus equation (4) reduces to a_{λ,μ}(x) = b_{μ,λ}(θ(x))J(x).

Meanwhile, the change of variables theorem in ℝ^{n} shows that

\begin{aligned}∫ ⋯∫ b_{μ,λ}(y) dy_{1} dy_{2} … dy_{n} & = ∫ ⋯∫ b_{μ,λ}(θ(x))J(x) dx_{1} dx_{2} … dx_{n} \\ & = ∫ ⋯∫ a_{λ,μ}(x) dx_{1} dx_{2} … dx_{n} \end{aligned}

as required; since θ preserves the orientation of ℝ^{n}, J > 0 and we do not need to write |J(x)| here.

Theorem 5 Suppose θ: P ≅ M is an orientation-preserving diffeomorphism of oriented smooth n-manifolds, and that ω is a continuous n-form on M. Then ∫_{P} θ^{∗}ω = ∫_{M} ω.

Proof Take a smooth atlas {h_{λ}: U_{λ} ≅ U′_{λ} ⊂ ℝ^{n} : λ ∈ Λ} with partition of unity {ϕ_{λ} : λ ∈ Λ} as above. There is an obvious atlas on P, consisting of open sets W_{λ} = θ^{−1}(U_{λ}) with local charts

W_{λ} − ; − ; →^{θ} U_{λ} − ; − ; →^{h_{λ}} U′_{λ} ⊂ ℝ^{n},

and the functions ϕ_{λ} ^{∘} θ = θ^{∗}ϕ_{λ} form a partition of unity on P. The pullback of (θ^{∗}ϕ_{λ})(θ^{∗}ω) on W_{λ} to U′_{λ} is

((h_{λ}^{∘}θ)^{−1})^{∗}((θ^{∗}ϕ_{λ})(θ^{∗}ω)) = (h_{λ}^{−1})^{∗}(θ^{−1})^{∗}θ^{∗}(ϕ_{λ}ω) = (h_{λ}^{−1})^{∗}(ϕ_{λ}ω).

Our choices make ∫_{P} θ^{∗}ω and ∫_{M} ω identical, term by term.

Integration on a Lie group From now on, we assume that G is a compact Lie group of dimension n. We wish to integrate a continuous real-valued function f over G. We write the integral as ∫_{G} f(g) dg, where g denotes the variable of integration and dg should be considered a measure (instead of any kind of differential form) on G, known as Haar measure. (However, we avoid doing any measure theory.)

Denote by C(G) the vector space of continuous real-valued functions on G. The integral is required to have the following four properties:

(i) It is an ℝ-linear function C(G) → ℝ; (ii) It is monotone: if f(g) ≥ 0 for all g, then ∫_{G} f(g) dg ≥ 0; (iii) It is left-invariant: for any h ∈ G, ∫_{G} f(hg) dg = ∫_{G} f(g) dg; (iv) It is normalized: ∫_{G} 1 dg = 1. (6)

Theorem 7 Given a compact Lie group G, there exists a unique integral C(G) → ℝ that satisfies the axioms (6).

To prove existence, we treat G as an n-manifold and apply the preceding theory of integration of n-forms. (Uniqueness will be included in Corollary 13.)

First, we need to orient G, which is easy. We choose an orientation o_{e} of the tangent space T_{e}(G). We use the left translations l_{h} to propagate this to an orientation on the whole of G: given any basis {v_{1}, v_{2}, …, v_{n}} of T_{h}(G), for any h ∈ G, we define

o_{h}(v_{1}, v_{2}, …, v_{n}) = o_{e}(T (l_{h^{−1}})v_{1}, T (l_{h^{−1}})v_{2}, …, T (l_{h^{−1}})v_{n}).

This orientation is left-invariant: for any basis {v_{1}, v_{2}, …, v_{n}} of T_{g}(G), we have

o_{hg}(T (l_{h})v_{1}, T (l_{h})v_{2}, …, T (l_{h})v_{n}) = o_{g}(v_{1}, v_{2}, …, v_{n}),

and it is the only left-invariant orientation that extends o_{e}.

Next, we need an n-form on G. We choose a positively oriented alternating nform ω_{e} on the vector space T_{e}(G), and similarly propagate it around G by taking ω_{h} = l_{h^{−1}}^{∗}ω_{e} for all h ∈ G, so that ω is left-invariant, l_{h}^{∗}ω = ω for all h ∈ G.

Definition 8 We define ∫_{G} f(g) dg = ∫_{G} f mω, where the number m is independent of f and is chosen to make (6)(iv) hold.

The axioms (6) are readily verified. Axiom (i) is clear. By the Remark following Definition 2, since ω is positively oriented everywhere, ∫ ω > 0 and m > 0 can indeed be chosen to make axiom (iv) hold; further, axiom (ii) holds because ∫ f mω ≥ 0 if f ≥ 0 everywhere. For axiom (iii), we use l_{h}^{∗}ω = ω and Theorem 5 to express the left invariance as

∫ f^{∘}l_{h} mω = ∫ l_{h}^{∗}f mω = ∫ l_{h}^{∗}f ml_{h}^{∗}ω = ∫ l_{h}^{∗}(f mω) = ∫ f mω.

Remark If we reverse the orientation of G, we change ω to −ω, but m stays the same. By the second Remark after Definition 2, the integral ∫ f(g) dg does not change.

It is easy to construct a right-invariant integral, by introducing an inverse. It apparently requires a different measure, denoted by δg instead of dg.

Definition 9 We define the right-invariant integral

∫_{G} f(g) δg = ∫_{G} f(g^{−1}) dg.  (10)

This clearly is right-invariant,

∫ f(gh) δg = ∫ f(h^{−1}g^{−1}) dg = ∫ f(g^{−1}) dg = ∫ f(g) δg.

The other three axioms are obvious.

We shall see later, in Corollary 13, that the two integrals in fact coincide, so that δg = dg.

To prove this, we need a Fubini-type result, which has nothing to do with Lie groups, or even manifolds. Let X and Y be compact metric spaces. Suppose we are given “integrals” I_{X}: C(X) → ℝ and I_{Y}: C(Y) → ℝ that satisfy axioms (i) and (ii) of (6). We define a partial integration over Y, Î_{Y}: C(X ×Y) → C(X): given F: X × Y → ℝ, we define Î_{Y} F ∈ C(X) by (Î_{Y} F)(x) = I_{Y} F_{x}, where F_{x} ∈ C(Y) is given by F_{x}(y) = F (x, y). We similarly define Î_{X}: C(X ×Y) → C(Y).

Lemma 11 For any F ∈ C(X ×Y), I_{X} Î_{Y} F = I_{Y} Î_{X}F.

Proof We observe that if we equip the spaces C(X) etc. with the sup norm, the hypotheses imply that I_{X} and Î_{X} become bounded linear operators with norm vol(X), and I_{Y} and Î_{Y} have norm vol(Y), where we introduce the volumes vol(X) = I_{X}1 and vol(Y) = I_{Y} 1.

By compactness, F is uniformly continuous. Given ϵ > 0, choose δ > 0 such that d(x_{1}, x_{2}) < δ and d(y_{1}, y_{2}) < δ imply that |F (x_{2}, y_{2}) − F (x_{1}, y_{1})| < ϵ. If d(x_{1}, x_{2}) < δ, ∥F_{x_{2}} − F_{x_{1}}∥ < ϵ and |(Î_{Y} F)(x_{2}) − (Î_{Y} F)(x_{1})| < ϵ vol(X), which shows that Î_{Y} F does indeed lie in C(X).

The result is obvious for a function of the form F (x, y) = ϕ(x)ψ(y); both sides reduce by ℝ-linearity to (I_{X}ϕ)(I_{Y} ψ). For a general F, we approximate by sums of such functions. We cover X by finitely many open sets U_{λ} of diameter less than δ, take a partition of unity consisting of functions ϕ_{λ} with support in U_{λ}, and choose points x_{λ} ∈ U_{λ}; similarly, we cover Y by open sets V_{μ} with functions ψ_{μ} and points y_{μ}. We may then write

F (x, y) = ∑_{λ} ∑_{μ} ϕ_{λ}(x)ψ_{μ}(y)F (x, y).

We approximate F by the function

E(x, y) = ∑_{λ} ∑_{μ} ϕ_{λ}(x)ψ_{μ}(y)F (x_{λ}, y_{μ});

then

|F (x, y) − E(x, y)| ≤ ∑_{λ} ∑_{μ} ϕ_{λ}(x)ψ_{μ}(y) |F (x, y) − F (x_{λ}, y_{μ})|.

For ϕ_{λ}(x) and ψ_{μ}(y) to be both nonzero, we must have x ∈ U_{λ} and y ∈ V_{μ}, hence |F (x_{λ}, y_{μ}) − F (x, y)| < ϵ. This yields, for all x and y,

|F (x, y) − E(x, y)| ≤ ∑_{λ} ∑_{μ} ϕ_{λ}(x)ψ_{μ}(y)ϵ = ϵ.

Thus ∥F − E∥ ≤ ϵ, which implies that |I_{X} Î_{Y} F − I_{X} Î_{Y} E| ≤ ϵ vol(X)vol(Y) and |I_{Y} Î_{X}F − I_{Y} Î_{X}E| ≤ ϵ vol(X)vol(Y). But E is an ℝ-linear combination of the functions ϕ_{λ}(x)ψ_{μ}(y), for which we have I_{X} Î_{Y} E = I_{Y} Î_{X}E. Then |I_{X} Î_{Y} F − I_{Y} Î_{X}F | ≤ 2ϵ vol(X)vol(Y). Since ϵ is arbitrary, we must have I_{X} Î_{Y} F = I_{Y} Î_{X}F.

We apply this to the case X = Y = G.

Theorem 12 Let I_{Y} be any integral on G that satisfies axioms (6), in particular, left invariance, and I_{X} be any right-invariant integral on G that satisfies the axioms (with (iii) modified). Then I_{Y} = I_{X}.

Corollary 13 There are exactly one left-invariant integral and one right-invariant integral on G, and they are equal.

Remark Dealing directly with n-forms and orientations is more complicated and is best avoided. A left-invariant n-form ω on G is not right-invariant in general, unless G is connected; instead, the n-form ν^{∗}ω, where ν: G → G denotes inversion, is right-invariant, but satisfies ν^{∗}ω = ±ω, where the sign can vary from component to component of G.

Orientations behave the same way.

Proof of Theorem Given f ∈ C(G), we consider F (g, h) = f(gh). We note that F_{g}(h) = f(gh) = (f^{∘}l_{g})(h), or F_{g} = f ^{∘} l_{g}. Integrating over h, we have

(Î_{Y} F)(g) = I_{Y} F_{g} = I_{Y} (f^{∘}l_{g}) = I_{Y} f,

which is independent of g. Then I_{X} Î_{Y} F = I_{X}(I_{Y} f.1) = (I_{Y} f)(I_{X}1) = I_{Y} f. Similarly, I_{Y} Î_{X}F = I_{X}f.

Theorem 14 Let G be a compact Lie group and f ∈ C(G). Then for any h ∈ G and any automorphism ϕ of G, we have

∫ f(g) dg = ∫ f(hg) dg = ∫ f(gh) dg = ∫ f(g^{−1}) dg = ∫ f(ϕ(g)) dg.

Proof The first two integrals are equal by left invariance. The fourth is our definition of the right-invariant integral, which we have just seen is equal to the first. So the left-invariant integral is also right-invariant, which gives the third integral.

For the fifth, a different strategy is needed. We define the integral If = ∫ f(ϕ(g)) dg. It clearly satisfies axioms (i), (ii) and (iv). For (iii), we have

I(f^{∘}l_{h}) = ∫ f(hϕ(g)) dg = ∫ f(ϕ(ϕ^{−1}(h)g)) dg = ∫ (f^{∘}ϕ)(ϕ^{−1}(h)g) dg
= ∫ (f^{∘}ϕ)(g) dg by left invariance, (6)(iii)
= ∫ f(ϕ(g)) dg = If.

By uniqueness, If must coincide with the first integral, ∫ f(g) dg.