The Exterior Derivative and de Rham Complex

Posted on 2025-06-19 In Math

Outline

Chapter II: Differential Operators

6 The Exterior Derivative; de Rham Complex

6.1. Denition. Let α be a dierential form of degree p. Then we dene the exterior

derivative dα of α to be the dierential form of degree p + 1 given by the formula

(dα)(X

, . . . , X

p+1

) =

p+1



i=1

(−1)

i+1

α(X

, . . . ,

, . . . , X

p+1

)



1≤i<j≤p+1

(−1)

i+j

α([X

, X

], X

, . . . ,

, . . . , X

p+1

)

where a hat over a symbol means that the symbol does not occur.

6.2. Remark. Firstly, one easily checks that dα is A-linear. To show that it is

alternating, notice that if X

= X

= X, k < l, the terms of the rst type on the right

side of the denition reduce to two terms, namely those corresponding to i = k and

i = l, and it is clear that these cancel out. On the other hand, in the second type of

terms, we have to consider the summation over terms with i = k, j 6= l and i 6= k, j = l,

while the term corresponding to i = k, j = l is zero as it involves [X

, X

] in one of the

arguments. It is easy to see that the above two terms also cancel out, proving that dα

is indeed alternating.

6.3. Example. If α is a form of degree 0, namely a function f, then the denition

gives df(X) = Xf. If α is of degree 1, then dα(X, Y ) = Xα(Y ) −Y α(X) − α([X, Y ]).

6.4. Denition. If X is a vector eld and α is a dierential form of degree p, we

dene the inner product ι

α of α with respect to X to be a form of degree p −1 given

(ι

α)(X

, . . . , X

p−1

) = α(X, X

, . . . , X

p−1

6.5. Lemma. For forms α, β of degree p, q we have

(α ∧ β) = ι

α ∧ β + (−1)

α ∧ ι

(β).

Proof. This is a simple consequence of Denitions 6.1 and 6.4.

6.6. Remarks.

1) The space of all dierential forms constitutes a graded associative algebra. It

has also a Z/2-gradation coming from its gradation. If a linear endomorphism

preserves the parity of forms, we say that it is Z/2-homogeneous of even type.

Similarly if an endomorphism takes odd forms to even ones, and even to odd,

we say it is homogeneous of odd type. A linear endomorphism is said to be a

Section 6: The Exterior Derivative; de Rham Complex

derivation of even type if it is homogeneous of even degree and is a derivation in

the usual sense. But if an endomorphism D is homogeneous of odd degree and

satises D(α ∧ β) = Dα ∧ β + (−1)

deg α

α ∧ Dβ, we say that it is a derivation of

odd type. For example, by Lemma 6.5, the inner product ι

is a derivation of odd

type while L(X) is one of even type. If both of them are of even type, we have

seen that D

−D

is also a derivation of even type. If one of them is of odd

type and the other even, then D

−D

is still a derivation, but of odd type.

Finally if both of them are odd derivations, then D

+ D

is a derivation of

even type.

2) In terms of the linear map ι

, formula 5.10 can be rewritten as

L(fX)(α) = f L(X)(α) + df ∧ ι

for a 1-form α. Using the fact that L(f X), fL(X) and the map α 7→ df ∧ι

α are

all derivations, one sees that this formula is valid for forms of arbitrary degree. In

particular, if α is of degree n, we have 0 = ι

(df ∧ α) = ι

(df) ∧ α − df ∧ ι

α =

(Xf )α − df ∧ ι

α, so that L(f X)(α) = f L(X)α + (Xf )(α).

6.7. Proposition. The exterior derivative d has the following properties.

i) It is additive, namely, d(α + β) = dα + dβ.

ii) For any vector eld X, we have dι

+ ι

d = L(X) on dierential forms.

iii) If α and β are dierential forms of degree p and q, respectively, then

d(α ∧ β) = dα ∧ β + (−1)

α ∧ dβ.

In other words, d is a derivation of odd type.

iv) d ◦ d = 0.

Proof. i) is obvious from the denition of d.

ii) In fact, we have

(dι

+ ι

d)(α)(X

, . . . , X

)



(−1)

i+1

(ι

α)(X

, . . . ,

, . . . , X

)



(−1)

i+j

(ι

α)([X

, X

], . . . ,

, . . . ,

, . . . X

) + (dα)(X, X

, . . . , X

)

= Xα(X

, . . . , X

) +



(−1)

α([X, X

], X

, . . . ,

, . . . , X

)

= (L(X)α)(X

, . . . , X

)

for any form α of degree p and vector elds X

, . . . , X

Chapter II: Differential Operators

iii) is proved by induction on p + q. Our assertion is obvious from the denition if

one of p, q is 0. So let us assume that p + q > 0. In order to prove the equality

of the dierential forms on the two sides, it is enough to show that they are the

same after applying ι

for any arbitrary X. Now we may use ii) and the induction

assumption to complete the proof.

iv) Since d is a derivation of odd type, we have, by Remark 6.6, 1), that d◦d+d◦d = 2d

is an even derivation. In order to show that it is 0, it is enough to check this on

the generators of the R-algebra of dierential forms, namely functions and 1-forms.

Firstly, (d

f)(X, Y ) = X(df)(Y ) − Y (df)(X) − df([X, Y ]) = XY (f ) − Y X(f ) −

[X, Y ](f ) = 0 . Using the denition we may check similarly that it is 0 also on

1-forms. Alternatively, we note that it is enough to prove that it is 0 on small

open sets. In a domain in R

, however, it is clear that any 1-form is a linear

combination of forms of the type α = f(x)dx. But then dα = df ∧ dx. It follows

therefore that d

vanishes on α.

6.8. Local computation. We have seen that on a domain U in R

, the operators

∂

∂x

, . . . ,

∂

∂x

generate the space of all vector elds as an A(U)-module. The dual basis

for dierential forms of degree 1 is (dx

, . . . , dx

) where (x

, . . . , x

) are the coordinate

functions and dx

are their exterior derivatives. For, if f is a function and X a vector

eld, we have by denition, df(X) = Xf. In particular, (dx

)



∂

∂x



∂x

= δ

. Hence

every dierential 1-form can be uniquely expressed as α =



. If f is any function,

we have by denition, (df)



∂

∂x



∂f

∂x

. Hence df =



∂f

∂x

. More generally, any

r-dierential form can be expressed as ω =



<···<i

,...,i

∧ ··· ∧ dx

, where

,...,i

are functions. Using 6.7 we obtain

dω = da

,...,i

∧ dx

∧ ··· ∧ dx

and since da

,...,i

has been computed above, this completes the local computation of

the exterior derivative. Moreover, this also gives the computation of L(X) for any

vector eld in local coordinates. One notes that these local expressions are rst order

dierential operators, in the sense that the eect of d on α involves the derivatives of

the coecient functions.

6.9. A digression on complexes. We recall the notions of complexes, morphisms

of complexes, homotopy between morphisms, … of A-modules, where A is any ring.

Analogous notions are available for sheaves and will be dealt with in Chapter 4, but we

need the basic denitions here. A complex of A-modules is a sequence M

of modules

and homomorphisms M

→ M

i+1

for all i ∈ Z such that the successive composites

i−1

→ M

i+1

are all 0. The homomorphisms are called the dierentials of the

Section 6: The Exterior Derivative; de Rham Complex

complex, and usually all of them are denoted by d. The complex itself is often denoted

by M

◦

6.10. Examples.

1) Recall that a singular r-cochain of a topological space is simply an assignment to

every singular r-simplex, of an element of a xed abelian group A. For every 0 ≤

i ≤ r, consider the restriction ∂

to the standard r + 1-simplex, of the linear map

r+1

→ R

r+2

dened on the elements of the standard basis as follows: ∂

) = e

for 0 ≤ j ≤ i and ∂

) = e

j+1

for i + 1 ≤ j ≤ r + 1. This gives a map

∆

→ ∆

r+1

. The i-th face of a singular r + 1-simplex s : ∆

r+1

→ X is dened

to be the composite s ◦ ∂

. Then we dene the dierential of a singular i-cochain

c by setting dc(s) =



(−1)

c(s ◦ ∂

). Then one can check that d

= 0 and the

resulting complex is called the singular complex of X, with coecients in A.

2) We have checked above that the exterior derivative on dierential forms gives rise

to a complex called the de Rham complex.

If M

◦

is a complex, one denes its cohomology groups H

to be the groups given

where Z

is the kernel of the map M

→ M

i+1

and B

is the image of the map

i−1

→ M

. Notice that since M

◦

is a complex, we have the inclusion B

⊂ Z

. Thus

can be taken as a measure of deviation of B

from being Z

. Another way of saying

it is that H

= 0 if and only if M

i−1

→ M

i+1

is exact, that is to say, the kernel

of M

→ M

i+1

is the same as the image of M

i−1

→ M

6.11. Denition. The cohomology of the singular complex with coecients in an

abelian group A is called the singular cohomology of the space with coecients in A.

A morphism f

◦

of a complex M

◦

into another complex N

◦

is a sequence of homo-

morphisms f

: M

→ N

for all i such that the diagrams

i−1

are all commutative. The maps (f

) induce in an obvious way homomorphisms Z

◦

) →

◦

) and B

◦

) → B

◦

) and consequently a homomorphism H

◦

) → H

◦

)

of the cohomology groups.

6.12. Remark. A continuous map f : X → Y of topological spaces gives in an

obvious way a morphism of the singular complex of Y into that of X. In fact, any

singular r-simplex s : ∆ → X in X gives rise to one in Y , namely f ◦ s. In particular,

it gives rise to a homomorphism of the singular cohomology group of Y into that of X.

Two morphisms f

◦

, g

◦

of complexes from M

◦

into N

◦

are said to be homotopic if

Chapter II: Differential Operators

there exist homomorphisms k

: M

→ N

i−1

for all i such that

+ k

i+1

d = f

− g

If f

◦

, g

◦

: M

◦

→ N

◦

are two morphisms of complexes which are homotopic, then

the induced homomorphisms on the cohomology groups are the same. This is trivial to

verify from the denition of homotopy. In particular, if M

◦

and N

◦

are homotopically

equivalent in the sense that there are morphisms in both directions such that both

composites are homotopic, then their cohomology groups are isomorphic.

6.13. Denition. The complex

0 → A → T

∗

→ Λ

∗

→ ···

with all sheaf homomorphisms given by the exterior derivative, is called the de Rham

complex.

6.14. Proposition. The de Rham complex is exact as a sequence of sheaves except

at the initial stage, and the kernel of A → T

∗

is the constant sheaf R.

Proof. Clearly it is enough to show that if U is an open cube in R

, then any form

ω =



<···<i

,...,i

∧ ··· ∧ dx

on U satisfying dω = 0 can be written as dα where α is an (r − 1)-form. We will

generalise this a little in order to set up a suitable induction. The generalisation we

intend is to let a

,...,i

to be not merely dierentiable functions on U , but dierentiable

functions on V × U where V is a parameter space assumed to be an open cube in R

Then we claim that the form α can be chosen to depend dierentiably on the parameter

space as well, in the sense that the coecient functions are also dierentiable on V ×U.

When we take a form in n variables with coecients which depend on m other variables

as well, the notation d for exterior derivative could be somewhat ambiguous. To obviate

this, we will denote by d

the exterior derivative when α is treated as a form on an open

set of R

, perhaps depending on some parameters. Now we want to use induction on n.

Any form as above (abbreviating the parameters to t) is best written as

ω(t) = ω

(t, x

) ∧ dx

+ ω

(t, x

)

where ω

(t, x

) and ω

(t, x

) are uniquely determined forms of degrees r − 1 and r,

respectively, which depend dierentiably on (t, x

) and do not involve dx

in the wedge

part of their expressions. In other words, they are dierential forms in n − 1 variables

Section 6: The Exterior Derivative; de Rham Complex

depending on m + 1 parameters. Then we get

ω(t) = d

(t, x

) ∧ dx

+ d

(t, x

)

= d

n−1

(t, x

) ∧ dx

+ d

n−1

(t, x

) + (−1)

∂ω

(t, x

)

∂x

∧ dx

Here when we write

∂ω

∂x

, we mean that we dierentiate all the coecient functions in

the local expression for ω. Thus we deduce that

ω(t) =



n−1

(t, x

) + (−1)

∂ω

(t, x

)

∂x



∧ dx

+ d

n−1

(t, x

But we are given that d

ω(t) = 0, so that we obtain two equations:







n−1

(t, x

) + (−1)

∂ω

(t,x

)

∂x

= 0,

n−1

(t, x

) = 0.

The second equation implies (by the induction assumption) that there exists a form

α(t, x

) in n − 1 variables such that d

n−1

α(t, x

) = ω

(t, x

). Then the rst equation

gives on substitution,

n−1

(t, x

) + (−1)

∂

∂x

n−1

α(t, x

) = 0,

or what is the same,

n−1



(t, x

) + (−1)

∂

∂x

α(t, x

)



= 0.

Again by the induction assumption, there exists a form β(t, x

) in n − 1 variables such

that

n−1

β( t, x

) = ω

(t, x

) + (−1)

∂

∂x

α(t, x

Now let us set γ(t) = β(t, x

) ∧ dx

+ α(t, x

). Then we have

γ(t) = d

n−1

β( t, x

) ∧ dx

+ d

n−1

α(t, x

) + (−1)

r−1

∂α(t, x

)

∂x

∧ dx



(t, x

) + (−1)

∂α(t, x

)

∂x



∧ dx

+ ω

(t, x

)

+ (−1)

r−1

∂α(t, x

)

∂x

∧ dx

= ω( t) ,

as was to be proved. In order to start the induction, we have to prove the assertion in

the case n = 1. This means that if f is a function of x, say in the interval (−1, +1), and

Chapter II: Differential Operators

parameters t, then there exists a function φ(x, t) such that

∂φ(x,t)

∂x

= f (x, t). Indeed the

function dened by φ(x, t) =



f(x, t)dx is clearly dierentiable in x and t and has the

required property.

Finally it is obvious that on a connected open set of R

, if df = 0 , i.e.

∂f

∂x

= 0 for

all i, then f is a constant.

6.15. Remark. While the de Rham complex is exact as a sequence of sheaves, it is

not true (in general) that the sequence

0 → R → A(M ) → T

∗

(M) → ···

is exact. In other words, it is not in general true that if a dierential form ω satises

dω = 0 (namely, a closed form) then it can be expressed as dα (namely, an exact form)

on an arbitrary manifold M.

6.16. Example. Let us consider the manifold S

. The dierential form ω = XdY −

Y dX where X, Y are restrictions of the coordinate functions on R

to S

, is clearly

closed since there are no nonzero forms of degree 2 on S

. But there is no function f

on S

such that df = XdY − Y dX. For, assume that such a function exists. Locally

on S

we can nd a function θ such that X = cos θ and Y = sin θ. Substituting in the

dening equation for ω, we see that ω = dθ. Since by assumption df = ω as well, we see

that d(f −θ) = 0 and so f and θ dier by a constant. In particular the local function θ

can be extended to the whole of S

. It is well known and easy to prove that this is not

possible.

The above example has to do with the fact that S

is not simply connected. This

suggests that the problem of the exactness of the global de Rham complex is related

to the topology of the manifold in question. We will take up this question for detailed

discussion in Chapter 4. For now, we simply wish to point to the phenomenon of an

exact sequence of sheaves

0 → F

′

→ F → F

′′

→ 0

not giving rise to an exact sequence

0 → F

′

(M) → F(M ) → F

′′

(M) → 0.

The study of this question leads to the concept of cohomology of sheaves which will be

studied in Chapter 4.