LECTURE 22: INTEGRATION ON ANIFOLDS 1. Orientations Let be a smooth manifold of dimension n, and let ω Ω n () be a smooth n-form. We want to define the integral ω. First assume = R n. In calculus we learned how to define integrals of multi-variable functions R n f(x)dx 1 dx n. If ϕ : R n R n is a diffeomorphism, then we have the change of variable formula: (x = ϕ(y)) (1) f(x) dx 1 dx n = R n f(ϕ(y)) det(dϕ) y dy 1 dy n. R n Now suppose ω is a smooth n-form on R n. Using the coordinates {x 1,, x n }, we can write ω = f(x)dx 1 dx n, where f is a smooth function on R n. It is natural to define ( ) (2) = fdx 1 dx n := f(x)dx 1 dx n. R n ω As we mentioned last time: whenever we define something on manifolds using local charts, we need to check that the definition is independent of the choices of coordinate systems. But then we immediately run into trouble: if the new coordinates are just {y 1 = x 1, y 2 = x 2,, y n = x n }, then ω = f(y)dy 1 dy 2 dy n and thus our definition above would give us ω = f(y)dy 1 dy 2 dy n. So the naive definition (2) above is not well-defined, unless we throw away coordinate systems like { x 1, x 2, x 3,, x n }. It turns out that this is almost all we need to do to define integrals of differential forms on manifolds. To see this let s continue to study the case = R n, but with a general coordinate system {y 1,, y n } on R n. Then the coordinate change {y 1,, y n } {x 1,, x n } is given by a diffeomorphism ϕ : R n y R n x. We need to write ω in the y-coordinate system: 1
2 LECTURE 22: INTEGRATION ON ANIFOLDS Lemma 1.1. If ϕ : R n R n is a diffeomorphism and x = ϕ(y), then ϕ (dx 1 dx n ) = det(dϕ y )dy 1 dy n Proof. If we denote ϕ = (ϕ 1,, ϕ n ), then ϕ x i = x i ϕ = ϕ i. So ϕ (dx 1 dx n ) = dϕ 1 dϕ n. But since dϕ 1 dϕ n ( 1, y, n) y = det(dϕ y ), we conclude dϕ 1 dϕ n = det(dϕ y )dy 1 dy n. It follows that in the coordinate system {y 1,, y n }, the n-form ω can be written as ω = f(ϕ(y)) det(dϕ) y dy 1 dy n. So our naive definition (2) would give us ω = f(ϕ(y)) det(dϕ) y dy 1 dy n. In view of the change of variable formula (1), we see that this formula coincides with the formula (2) as long as we assume the coordinate-change diffeomorphism ϕ satisfies the condition det(dϕ) y > 0, y R n. Such diffeomorphisms are called orientation-preserving diffeomorphisms. In conclusion, the formula (2) for R n ω is independent of orientation-preserving ways of choosing coordinates. Now let s move to manifolds. In view of the above discussions, we define Definition 1.2. Let be a smooth manifold of dimension n. (1) Two charts (ϕ α, U α, V α ) and (ϕ β, U β, V β ) are orientation compatible if the transition map ϕ αβ = ϕ β ϕ 1 α satisfies det(dϕ αβ ) p > 0 for all p ϕ α (U α U β ). (2) An orientation of is an atlas A = {(ϕ α, U α, V α ) α Λ} whose charts are pairwise orientation compatible. (3) We say is orientable if it has an orientation. Remark. Let U be a chart with coordinates {x 1,, x n }. We use the notation U to represent the same coordinate chart U but with twisted coordinates { x 1, x 2,, x n }. Then U and U are not orientation compatible. Let Ũ be any other coordinate chart such that Ũ U is connected. Then either or Ũ and U are orientation compatible, Ũ and U are orientation compatible.
As a consequence, we immediately see LECTURE 22: INTEGRATION ON ANIFOLDS 3 Corollary 1.3. If is connected and orientable, then admits exactly two different orientations. Example. For the real projective space RP n, we have constructed an atlas consisting of n + 1 charts. We have calculated the transition map ϕ 1,n+1 and got ( ) 1 ϕ 1,n+1 (y 1,, y n ) = y, y1 n y,, yn 1. n y n A simple computation yields det (dϕ 1,n+1 ) = ( 1) n+1 1 (y n ) n+1. It follows that for the atlas we constructed, (ϕ 1, U 1, V 1 ) and (ϕ n+1, U n+1, V n+1 ) are orientation compatible if and only if n is odd. One can do the same computation for other pairs of charts. In fact, it is true that RP n is orientable if and only if n is odd. 2. Integrations of n-forms on smooth manifolds Now assume is a smooth orientable n-manifold and fix an orientation A on. Let ω be any smooth n-form on (not necessary a volume form). To define ω, we first assume that ω is supported in a coordinate chart {ϕ, U, V } which is consistent with A. In this case there is a function f(x 1,, x n ) supported in U such that As in the Euclidian case we define (3) ω := ω = f(x 1,, x n )dx 1 dx n. U V f(x 1,, x n )dx 1 dx n, where the right hand side is the Lebesgue integral on V R n. To integrate a general n-form ω Ω n () on, we take a locally finite cover {U α } of that are compatible with the orientation A. Let {ρ α } be a partition of unity subordinate to {U α }. Now since each ρ α is supported in U α, each ρ α ω is supported U α also. We define (4) ω := ρ α ω. α U α We say that ω is integrable if the right hand side converges. One need to check that the definition (4) above is independent of the choices of orientationcompatible coordinate charts, and is independent of the choices of partition of unity. Theorem 2.1. The expression (4) is independent of the choices of {U α } and the choices of {ρ α }. Proof. We first show that (3) is well-defined. The argument is essentially the same as in the Euclidian case: if ω is supported in U, and if {x i α} and {x i β } are two orientation-compatible coordinate systems on U, so that ω = f α dx 1 α dx n α = f β dx 1 β dx n β,
4 LECTURE 22: INTEGRATION ON ANIFOLDS then we want to prove f α dx 1 α dx n α = f β dx 1 β dx n β. V α V β This is true, because dx 1 β dx n β = det(dϕ αβ )dx 1 α dx n α implies f α = det(dϕ αβ )f β. Since det(dϕ αβ ) > 0, the conclusion follows from the change of variable formula in R n. To prove (4) is well-defined, we suppose {U α } and {U β } are two locally finite cover of consisting of orientation-compatible charts, and {ρ α } and {ρ β } are partitions of unity subordinate to {U α } and {U β } respectively. Then {U α U β } is a new locally finite cover of, and {ρ α ρ β } is a partition of unity subordinate to this new cover. It is enough to prove ρ α ω = ρ α ρ β ω. α U α α,β U α U β This is true because for each fixed α, ρ α ω = ( U α U α β ρ β )ρ α ω = β U α U β ρ β ρ α ω. Finally we extend the change of variable formula form R n to manifolds. Definition 2.2. Let, N be orientable smooth n-manifolds, with orientations A and B respectively. A diffeomorphism ϕ : N is said to be orientation-preserving if for each (ψ β, X β, Y β ) B, the chart (ψ β ϕ, ϕ 1 (X β ), Y β ) on is compatible with A. Suppose, N are connected. It is easy to see that a diffeomorphism ϕ : N is orientation-preserving if and only if there exists one chart (ψ β, X β, Y β ) B, such that the chart (ψ β ϕ, ϕ 1 (X β ), Y β ) on is compatible with A. Similarly if there exists one chart (ψ β, X β, Y β ) B, such that the chart (ψ β ϕ, ϕ 1 (X β ), Y β ) on is incompatible with A, then for every chart (ψ β, X β, Y β ) B, the chart (ψ β ϕ, ϕ 1 (X β ), Y β ) on is incompatible with A. In this case we say ϕ is orientation-reverting. Now we state: Theorem 2.3 (The change of variable formula.). Suppose, N are n-dimensional orientable smooth manifolds, and ϕ : N is a diffeomorphism. (1) If ϕ is an orientation-preserving, then f ω = (2) If ϕ is an orientation-reverting, then f ω = N N ω. ω.
LECTURE 22: INTEGRATION ON ANIFOLDS 5 Proof. It is enough to prove this in local charts, in which case this is merely the change of variable formula in R n. Remark. If ω is a k-form on, where k < n = dim, then one cannot integrate ω over. However, for any k-dimensional orientable submanifold X, one can define ω by setting it X to be X ι ω, where ι : X is the inclusion map. Remark. If is not orientable, one cannot define integrals of differential forms as above. However, we can still integrate via densities.