Exterior Derivatives • In this section we define a natural differential operator on smooth forms, called the exterior derivative. It is a generalization of the diffeential of a function. Motivations: • Recall that not all 1-forms are differentials of functions: Given a smooth 1-form ω, a necessary condition for the existence of a smooth function f such that ω = df is that ω be closed, which means that it satisfies ∂ω ∂ω (1) j − i =0 ∂xi ∂xj in every smooth coordinate system. Proposition 1. Let ω be a smooth covector field. If ω satisfies (1) in some smooth chart around every point, then it is closed. Proof. Let (U, (xi)) be an arbitrary smooth chart. For each point p ∈ U, the hypothesis guarantees that there are some smooth coordinates (xj ) defined near p in which the analogue of (1) holds. We have e k k ∂ωi ∂ωj ∂ ∂x k ∂ ∂x k − = e ω − e ω ∂xj ∂xi ∂xj ∂xi e ∂xj ∂xj e 2 k k k 2 k k k  ∂ x k ∂x ∂ω   ∂ x k ∂x ∂x  = e ω + e e − e ω + e e ∂xi∂xj e ∂xi ∂xi ∂xi∂xj e ∂xj ∂xi 2 k k ` k 2 k k ` k  ∂ x k ∂x ∂x ∂ω   ∂ x k ∂x ∂x ∂x  = e ω + e e e − e ω − e e e ∂xi∂xj e ∂xi ∂xj ∂x` ∂xi∂xj e ∂xj ∂xj ∂x` 2 k 2 k e k ` k k  ∂ x ∂ x  k ∂x ∂x ∂x ∂ω  = e − e ω + e e e − e ∂xi∂xj ∂xi∂xj e ∂xi ∂xj ∂x` ∂x` =0 + 0.  e

— By Proposition 1, being a closed form is a coordinate-independent property, and thus one might hope to find a more invariant way to express it. — The key is that the expression in (1) is antisymmetric in the indices i and j, so it can be interpreted as the ij-component of an alternating tensor field, i.e. a 2-form. — We will define a 2-form dω by ∂ω ∂ω dω = X j − i dxi ∧ dxj , ∂xi ∂xj imanifold, we will show that there is a differential operator d : Ak(M) →Ak+1(M) satisfying d(dω) = 0 for all ω. — Thus it will follow that a necessary condition for a smooth k-form ω to be equal to dη for some (k − 1)-form η is that dω =0.

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• The definition of d in coordinates is straightforward:

0 J 0 J (2) d(X ωJ dx )=X dωJ ∧ dx , J J

where dωJ is just the differential of the function ωJ . — In somewhat more detail, this is

∂ωJ (3) d(X 0ω dxj1 ∧···∧dxjk )=X 0 X dxi ∧ dxj1 ∧···∧dxj+k. J ∂xi J J i — Observe that when ω is a 1-form, this becomes ∂ω d(ω dxj )= j dxi ∧ dxj j ∂xi ∂ω ∂ω = X j dxi ∧ dxj + X j dxi ∧ dxj ∂xi ∂xi ij ∂ω ∂ω = X j − i dxi ∧ dxj . ∂xi ∂xj imanifolds takes a little work. — This is the content of the next theorem. Theorem 2 (The Exterior Derivative). For every smooth manifold M, there are unique linear maps d : Ak(M) →Ak+1(M) defined for each k ≥ 0 and satisfying the following three conditions: (i) If f is a smooth, real-valued function (a 0-form), then df is the differential of f, defined as usual by df (X)=Xf. (ii) If ω ∈Ak(M) and η ∈A`(M), then

d(ω ∧ η)=dω ∧ η +(−1)kω ∧ dη. (iii) d ◦ d =0. This operator also satisfies the following properties: (a) In every smooth coordinate chart, d is given by (2). (b) d is local: If ω = ω0 on an open set U ⊂ M, then dω = dω0 on U. (c) d commutes with restrictions: If U ⊂ M is any open set, then

(4) d(ω )=(dω) . U U 3

Proof. (I) Begin with the special case: Suppose M can be covered by a single smooth chart. Let (x1, ··· ,xn) be global smooth coordinates on M, and define d : Ak(M) → Ak+1(M) by (2). — The map d thus defined is clearly linear and satisfies (i). (I.1) Claim: It satisfies (ii) and (iii). — Before doing so, we claim: d satisfies d(fdxI )=df ∧ dxI for any multi-index I, not just increasing ones; indeed, (1) if I has repeated indices, then clearly d(fdxI )=0=df ∧ dxI ; (2) if I has no repeated indices, then let σ be the permutation setting I to an increasing multi-index J, we have

d(fdxI ) = (sgn σ)d(fdxJ ) = (sgn σ)df ∧ dxJ = df ∧ dxI .

— To prove (ii), by linearity it suffices to consider terms of the form ω = fdxI and η = gdxJ . We compute

d(ω ∧ η)=d((fdxI ) ∧ (gdxJ )) =d(fgdxI ∧ dxJ ) =(gdf + fdg) ∧ dxI ∧ dxJ =(df ∧ dxI ) ∧ (gdxJ )+(−1)k(fdxI ) ∧ (dg ∧ dxJ ) =dω ∧ η +(−1)kω ∧ dη,

where the (−1)k comes from the fact that dg ∧ dxI =(−1)kdxI ∧ dg because dg is a 1-form and dxI is a k-form. — Prove (iii) first for the special case of a 0-form, i.e. a real-valued function. In this case,

∂f ∂2f j i ∧ j d(df )=d( j dx )= dx dx ∂x ∂xi∂xj ∂2f ∂2f = X − dxi ∧ dxj =0. ∂x ∂x ∂x ∂x i

For the general case, we use the k = 0 case together with (ii) to compute

 0 j1 jk  d(dω)=d X dωJ ∧ dx ∧···∧dx J

0 j1 jk = X d(dωJ ) ∧ dx ∧···∧dx J k 0 i j1 j1 jk + X X(−1) dωJ ∧ dx ∧···∧d(dx ) ∧···∧dx J i=1 =0.

This proves that there exists an operator d satisfying (i)-(iii) in the special case. 4

(I.2) Properties (a)-(c) are immediate consequences of the definition, once we note that if M is covered by a single smooth chart, then any subset of M has the same property. (I.3) To show that d is unique, suppose d˜ : Ak(M) →Ak+1(M) is another linear operator defined for each k ≥ 0 and satisfying (i), (ii) and (iii). 0 J — Let ω = PJ ωJ dx ∈A(M) be arbitrary. Using linearity of d˜ together with (ii), we compute

˜ ˜ 0 j1 jk dω =d(X ωJ dx ∧···∧dx ) J

0 ˜ j1 jk 0 ˜ j1 jk = X dωJ ∧ dx ∧···∧dx + X ωJ d(dx ∧···∧dx ). J J

– Using (ii) again, the last term expands into a sum of terms, each of which contains a factor of the form d˜(dxji ), which is equal to d˜(dx˜ ji ) by (i) and hence is zero by (iii). – On the other hand, since each component function ωJ is a smooth function, (i) implies that dω˜ J = dωJ . – Thus dω˜ is equal to dω defined by (2). – This implies, in particular, that we obtain the same operator no matter which (global) smooth coordinates we use to define it. – This completes the proof of the existence and uniqueness of d in the special case. (II) Next, let M be an arbitrarily smooth manifold. — On any smooth coordinate domain U ⊂ M, the argument above yields a unique linear operator from smooth k-forms to (k + 1)-forms, which we denote by dU , satisfying (i)-(iii). 0 — On any set U ∩ U where two smooth charts overlap, the restrictions of dU ω and 0 dU 0 ω to U ∩ U satisfy

(dU ω) = dU∩U 0 ω =(dU 0 ω) , by (4). 0 0 U∩U U∩U — Therefore, we can unambiguously define d : Ak(M) →Ak+1(M) by defining the value of dω at p ∈ M to be (dω)p = dU (ω|U )p, where U is any smooth coordinate domain containing p. — This operator satisfying (i), (ii), and (iii) because each dU does. – It also satisfies (a), (b), and (c) by definition. (II.1) Finally, we claim uniqueness in the general case. — Suppose we have some ther property

d : Ak(M) →Ak+1(M) e defined for eack k and satisfying (i)-(iii). (II.1.1) Begin by showing that d satisfies the locality property (b). e For this, writing η = ω − ω0, it suffices to

claim: dη =0on an open set U if η =0. e U 5

Indeed, for an arbitrary point p ∈ U, let ϕ ∈ C∞(M) be a smooth bump function that is equal to 1 in a neighborhood of 1 and supported in U. Then ϕη ≡ 0, and hence

0=d(ϕη) = dϕ ∧ η + ϕ(p)dη = dη , e p e p p e p e p because ϕ ≡ 1 in a neighborhood of p. Since p is an arbitrary point of U, this shows that dη =0onU. (II.1.2) Let U ⊂ M be an arbitrary smooth domain. For each k, define an operator

d : Ak(U) →Ak+1(U) eU as follows. For each p ∈ U, (1) choose an extension of ω to a smooth global k-form ω ∈Ak(M) that agrees with ω on a neighborhood of p, and e (2) set (dU ω)p =(dω)p. e ee Because d is local, this definition is independent of the extension ω chosen. e — The fact that d satisfies (i)-(iii) implies immediately that d satsfiese the same e eU properties. This implies that d = d , eU U by the uniqueness property proved in (I.3) for smooth coordinate domains. — In particular, if ω is the restriction to U of a global form ω on M, then we can use the same extension ω near each point, so e e

(dω) = dU (ω )=dU (ω )=(dω) . e U e U e e U ee U This shows that d is equal to the operator d we defined above.  e Definition. The operator d whose existence and uniqueness are asserted in theo- rem is called exterior differentiation, and dω is called the exterior derivative of ω. k Definition. If A = Lk A is a graded algebra, a linear map T : A → A is said to be of degree m if T (Ak) ⊂ Ak+m ∀k. • It is said to be antiderivative if it satisfies T (xy)=(Tx)y +(−1)kx(Ty) when- ever x ∈ Ak and y ∈ A`. Corollary. The exterior differential extends to a antiderivative of A∗(M) of degree 1 whose square is zero. 6

Definition. (1) A smooth differential form ω ∈Ak(M) is said to be closed if dω =0. (2) A smooth differential form ω ∈Ak(M) is said to be exact if ∃(k − 1)-form η on M such that ω = dη. Corollary. Every exact form is closed. Proof. This follows from d ◦ d =0. 

• One important feature of the exterior derivative is that it behaves well w.r.t. pullbacks, as the next lemma shows. Lemma 3 (Naturality of the Exterior Derivative).. If G : M → N is a smooth map, then the pullback map G∗ : Ak(N) →Ak(M) commutes with d:

(5) G∗(dω)=d(G∗ω), ∀ω ∈Ak(N).

Proof. Because d is local, it suffices, by linearity, to check (5) for a form of the type

fdxi1 ∧···∧dxik .

For such a form, the left-hand side of (5) is

G∗d(fdxi1 ∧···∧dxik )=G∗(df ∧ dxi1 ∧···∧dxik ) =d(f ◦ G) ∧ d(xi1 ◦ G) ∧···∧d(xik ◦ G),

while the right-hand side is

dG∗(fdxi1 ∧···∧dxik )=d((f ◦ G)d(xi1 ◦ G) ∧···∧d(xik ◦ G)) =d(f ◦ G) ∧ d(xi1 ◦ G) ∧···∧d(xik ◦ G).