7 Di erential forms: algebraic theory - School of Mathematicstv/Teaching/Differentiable...

Theodore Voronov. Differentiable Manifolds. Autumn 2014

§7 Differential forms: algebraic theory

Last updated: November 5, 2014.

The notion of a differential form is somewhat familiar from the course of calculus with manyvariables. If x1, . . . , xn are local coordinates, then a differential k-form is an expression of theform

ω = ωi1...ik dxi1 ∧ . . . ∧ dxik . (1)

The summation is over all combinations of indices. The coefficients ωi1...ik transform in such away so to make the whole expression invariant. They may be constants, then we speak aboutforms at a point, or functions, then we speak about forms on a manifold (or a domain of amanifold). The rules of multiplication (called the ‘exterior product’) are such that

dxi ∧ dxj = −dxj ∧ dxi , (2)

for all i and j. Forms are related with integration, so that 1-forms can be integrated over curves,2-forms can be integrated over 2-dimensional surfaces, and so on. On forms, there is an operatord increasing degrees, which extends the usual differential of functions.

Our goal is to develop theory of forms in a more systematic way in the context of differentiablemanifolds. We start with the algebraic theory.

7.1 One-forms

7.1.1 The dual of the tangent space

Example 7.1. Consider a smooth function on M , i.e., a map f : M → R. We identify tangentvectors to R with numbers. Hence the tangent map df(x) at a point x ∈ M is a linear mapTxM → R. In coordinates it is given by the matrix (row-vector)(

∂f

∂x1, . . . ,

∂f

∂xn

).

The value of df(x) on a tangent vector v with coordinates vi will be

df(x)(v) =∂f

∂x1v1 + . . .+

∂f

∂xnvn .

Thus we have recovered the classical understanding of the differential of a function as the main(linear) part of the increment of the function for a given increment of the independent variables.The value of df(x) on v is traditionally denoted as ∂vf and is called the derivative along v(sometimes also “directional derivative”, but it is not a good term, as it depends on a vector vand not just on its direction).

Remark 7.1. An identification of the differential of a function on Rn with a vector called‘gradient’ used in elementary calculus books relies on a choice of an inner product on Rn. Thevector grad f at a point x ∈ Rn is defined so that for all vectors w,

grad f ·w = df(x)(w) .

1


The dot here denotes an inner product (= scalar product). In particular, in the basis associatedwith the standard Cartesian coordinates on Rn, which is orthonormal for a ‘standard’ innerproduct, the components of grad f are the partial derivatives, i.e., coincide with the componentsof df . But this simple identification fails in non-Cartesian coordinates. On manifolds, the vectorgrad f does not make sense unless an extra structure is introduced. On the other hand, thedifferential df is an intrinsic object not requiring anything for its definition.

We may develop the previous example slightly to get the following. Recall that, for vectorspaces V and W , the set of all linear mappings V → W , denoted hom(V,W ), is a vector spaceitself (of dimension equal to the product of dimensions of V and W ). It is isomorphic to thespace of rectangular m×n matrices if dimV = n and dimW = m. In particular, all linear mapsV → R make a vector space of the same dimension as V .

Definition 7.1. The vector space of all linear maps V → R is called the dual space (to a vectorspace V ) and is denoted V ∗ . Its elements are called covectors.

So the differential of a function f : M → R at x ∈ M is an element of the dual space toTxM .

Definition 7.2. The space dual to the tangent space TxM is called the cotangent space at x .Notation:

T ∗xM := (TxM)∗ .

Its elements are called covectors at x. The union of all cotangent spaces is denoted T ∗M ,

T ∗M :=⋃x

T ∗xM ,

and called the cotangent bundle of M .

The cotangent bundle has a natural structure of a manifold of dimension 2n if dimM = n.It is another example of a vector bundle.

If we are given coordinates x1, . . . , xn near a point x ∈M , we have n covectors dx1, . . . , dxn

at x. For a tangent vector v ∈ TxM ,

dxi(v) = ∂vxi = vi

(the ith component of v) if v =∑viei. The differentials dxi make a basis of the cotangent

space T ∗xM . Every covector ξ ∈ T ∗xM acts on v ∈ TxM as

ξ(v) = ξ(∑

viei)

=∑

viξ(ei) =∑

viξi ,

where we have denoted ξi := ξ(ei). Therefore we can write that as

ξ =∑

ξidxi .

For the basis vectors ei = ∂x/∂xi, we have

dxi(ej) = ∂ej(xi) =

∂xi

∂xj=

1 for i = j

0 for i 6= j= δij

2


(the Kronecker symbol).For an arbitrary vector space V and a basis ei in V , a basis ei in V ∗ satisfying

ei(ej) =

1 for i = j

0 for i 6= j= δij

is called the dual basis for a basis ei.Therefore, the differentials of local coordinates dxi make a basis of the cotangent space T ∗xM

dual to the basis of partial derivatives ei = ∂x/∂xi in the tangent space TM .

Remark 7.2. We use different position of indices to distinguish between TxM and T ∗xM . Thecomponents of tangent vectors are denoted with upper indices, e.g., vi, and the basis vectors inTxM have lower indices: ei = ∂x

∂xi. The components of covectors are denoted with lower indices,

e.g., αi, and the basis covectors ei = dxi have upper indices. Therefore in the sums that haveinvariant meaning the summation is always over repeating indices, so that the summation indexrepeats once as an upper index and once as a lower index. Examples:

v =∑

viei

(the expansion of a vector over a basis);

α =∑

αidxi

(the expansion of a covector over the dual basis);

α(v) =∑

αivi

(the value of a covector on a vector, expressed in terms of the respective components).

Remark 7.3. A vector space V and its dual V ∗ have the same dimension and therefore are bothisomorphic to Rn, where n = dimV = dimV ∗. However, there is no way to identify V and V ∗

naturally. An attempt to identify a vector with components vi in some basis ei with the covectorhaving the same numbers as the components in the dual basis ei so that αi = vi fails because it doesnot survive a change of basis (so such an identification will depend essentially on a choice of basis). Wehave already mentioned that the partial derivatives ∂f/∂xi, which are the components of the differentialdf (a covector) can be interpreted as the components of a vector, the gradient grad f , as it is customaryin elementary expositions of multivariate calculus, only in fixed Cartesian coordinates. This naiveidentification survives only the linear changes of coordinates given by orthogonal matrices. In a deeperapproach, an identification of the dual space V ∗ with V is equivalent to a choice of a non-degeneratebilinear function V × V → R, for example, a scalar product1 on V . For TxM such as a choice of ascalar product smoothly depending on x, is called the Riemannian metric on a manifold M . Using aRiemannian metric one can identify tangent vectors with covectors and vice versa. On a Riemannianmanifold (that is, endowed with a Riemannian metric) the vector grad f makes perfect sense as thevector corresponding to the covector df . Its components are given by the formula (grad f)i =

∑gij∂jf

where gij are the matrix entries of the inverse matrix for the Gram matrix (gij), where gij = (ei, ej).

1A bilinear function B : V × V → R is called non-degenerate if for a fixed v ∈ V , the covector B(v,−) iszero only for v = 0, and the same for a fixed w and the covector B(−,w). A scalar product (v,w) = B(v,w) issingled out by the symmetry condition (v,w) = (w,v) and the positive-definiteness (v,v) > 0 for v 6= 0.

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Remark 7.4. Whilst there is no natural isomorphism between a vector space V and its dual V ∗ (apartfrom an identification using a particular basis or with the help of an extra structure such as a scalarproduct — both methods being not ‘natural’), there is a natural identification between the space V andthe second dual V ∗∗, i.e., the dual to the dual space V ∗. Indeed, for every vector v ∈ V , the value onv of an arbitrary covector α,

α(v) ,

is a linear function of α. Denote it v∗, so that v∗(α) = α(v). Hence there is a map V → V ∗∗, which isobviously linear. In coordinates, v∗(α) = viαi, from where we see that v 7→ v∗ is onto and injective, soit is indeed an isomorphism V → V ∗∗. In particular, the elements of a basis ei in V are mapped to theelements of the basis in V ∗∗ that is dual to the dual basis ei of V ∗ (i.e., dual to the original basis in V ).We see that it is possible to identify V and V ∗∗ and with such an identification the relation between abasis in V and the dual basis in V ∗ becomes entirely symmetric.

7.1.2 1-forms

Definition 7.3. A 1-form at a point x ∈M is the same thing as a covector at x, i.e., an elementα ∈ T ∗xM of the cotangent space. A 1-form α on a manifold M (or an open domain U ⊂M) isa smooth map M → T ∗M (or U → T ∗U ⊂ T ∗M) such that the value at each point x belongsprecisely to T ∗xM , the cotangent space at this particular point.

(Compare with the definition of vector fields, which are maps M → TM where the valuev(x) belongs to TxM . The standard terminology does not have different names for forms on Mor at a domain U ⊂M and forms at a point2, so each time we either have to say explicitly whatwe have in mind or this should be clear from the context.)

We shall systematically use the following equivalent description: a 1-form α on a manifoldM is a smooth function TM → R linear on tangent vectors.

Recall that TM =⋃TxM , so the linearity condition for α means that, for each x, the

function α(x,v) is a linear function TxM → R ; and α(x,v) depends smoothly on x ∈ M . So,for a fixed x, a 1-form at x is the same as a covector at x, and when x varies, it smoothlydepends on x.

We shall write α(x,v) = α(x)(v) separating the arguments in the notation, so that when xis fixed, we suppress it in the notation altogether and speak about α(v) as a function on tangentvectors.

For given coordinates x1, . . . , xn near x, there is a basis of the cotangent space T ∗xM consistingof the differentials dxi (see subsection 4.2.3 of §4). Hence a 1-form at x can be expanded overthis basis as

α = αidxi .

The numbers αi are called the components of a 1-form. If v ∈ TxM is a tangent vector at x andv = vi∂i, the value of α at v is given by

α(v) = αivi = viαi .

Example 7.2. We already know that the differential df of a function f ∈ C∞(M) at a givenpoint x ∈M is a 1-form at x :

df(x) =∂f

∂xi(x) dxi .

2Compare with ‘vector fields’ and ‘vectors’.

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Its value at a tangent vector v ∈ TxM is the derivative of f along v :

df(x)(v) =∂f

∂xi(x) vi = ∂vf .

Proposition 7.1. Under a change of coordinates, the components αi of a 1-form at a pointx ∈M transform according to the law

αi =∂xi

′

∂xiαi′ ,

where the partial derivatives are calculated at the argument (x1, . . . , xn) that corresponds to thepoint x.

Proof. Follows from the transformation law for dxi and the fact that the expression αidxi must

remain invariant under a change of coordinates:

αidxi = αi′dx

i′ = α .

Compare Proposition 7.1 with the transformation law for components of a vector.

Remark 7.5. One can see that, if written in the matrix form so that the components of vectorsand covectors are both arranged as column-vectors (or, both, as row-vectors), the transformationlaws for the components of vectors and covectors (1-forms) are given by the matrices that arerelated by the operation of taking the inverse and transpose. If the components of vectors arewritten as a column and the components of a covector (1-form) are written as a row, as it isfrequently done, the respective transformation laws will be given by the mutually reciprocalmatrices (without the transpose). This is most obviously seen from the formula

α(v) = αivi = (α1, . . . , αn)

v1

...vn

giving the expression of the value of a covector on a vector, in components.

If we allow the point x to vary, so that we have a covector in T ∗xM for each x, then we havea 1-form on M , whose coordinate expression in a coordinate domain U ⊂M looks as above,

α = αidxi = αi(x)dxi ,

with the only difference that the components αi are now smooth functions on U . They aretransformed by the same law as above,

αi =∂xi

′

∂xiαi′ ,

where now the partial derivatives are functions, not numbers.

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Example 7.3. The differential of a function f ∈ C∞(M) is a 1-form on M . On a coordinatedomain,

df =∂f

∂xidxi .

The differentials of functions seem to be the model examples of 1-forms, in the same way as thevelocities of curves are the model examples of tangent vectors. A question: which 1-forms canbe written as the differentials of functions? The answer is obvious for a fixed point: for a fixedx ∈ M , every 1-form α = αidx

i is the differential at this point of some function defined nearx. For arbitrary numbers αi, we can consider the linear function f(x) = αi x

i and obviouslyits differential df equals α = αi dx

i. However, if we consider 1-forms on M or on a domainU ⊂M allowing the point x ∈M to vary and the components αi to be functions, the situationdrastically changes. Indeed, if α = df , then in local coordinates

αi =∂f

∂xi,

and the partial derivatives satisfy the commutativity condition

∂2f

∂xi∂xj=

∂2f

∂xj∂xi,

so the components αi have to obey the relation

∂αj∂xi

=∂αi∂xj

or, in a more compact notation,∂iαj − ∂jαi = 0 .

This is a necessary condition for the existence of f such that α = df . It may be sufficient ormay be not. We shall come back to that.

7.2 Tensor and exterior products. The algebra of forms

Consider again objects defined at a fixed point x ∈ M . If we are given two 1-forms, α and β,at x, i.e., two linear functions of vectors in TxM , we can multiply them to obtain a function oftwo tangent vectors, α(v)β(w). Such a function on TxM × TxM is denoted with the symbol ⊗(pronounced ‘tensor’) and is called the tensor product of α and β:

(α⊗ β)(v,w) := α(v)β(w) .

Clearly, the function α ⊗ β is bilinear (i.e., linear in each argument), but does not, in general,satisfy any symmetry properties.

Example 7.4. For the differentials of local coordinates we have dxi ⊗ dxj, where

(dxi ⊗ dxj)(v,w) = dxi(v) dxj(w) = viwj .

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In the same way one can consider the tensor product of more than two 1-forms, the resultbeing a multilinear function of several tangent vectors. Such multilinear functions are alsoknown as tensors at x (more specifically, the covariant tensors). Since the value of every k-linear function on vectors v1, . . . , vk is a linear combination of various products vi11 . . . v

ikk of

their components, it follows that all possible k-fold tensor products of the differentials dxi makea basis in the space of the covariant k-tensors (i.e., the tensors with k vector arguments):

dxi1 ⊗ . . .⊗ dxik .

Example 7.5. For covariant 2-tensors, we have

T = Tij dxi ⊗ dxj ,

so that for arbitrary vectors v and w, the value of T on them is given by

T (v,w) = Tij viwj .

The tensor multiplication can be extended from 1-forms to arbitrary covariant tensors, sothat, in particular,(

dxi1 ⊗ . . .⊗ dxip)⊗(dxip+1 ⊗ . . .⊗ dxip+q

)= dxi1 ⊗ . . .⊗ dxip+q .

It is an associative, but not commutative operation.Many concrete applications require tensors satisfying particular symmetry properties.

Example 7.6. The coefficients of a quadratic form Q(v) = Qijvivj correspond to a symmetric

tensor,

S(v,w) := Qijviwj =

1

2

(Q(v + w)−Q(v)−Q(w)

).

We haveS(v,w) = S(w,v) .

In particular, for the components Sij = Qij we have the symmetry property Sij = Sji.

A Riemannian metric on a manifold is an example of a symmetric 2-tensor. Its value ontangent vectors v and w is their scalar product (v,w) w.r.t. this metric.

On the other hand, applications to integration require antisymmetric tensors, i.e., thosemultilinear functions of vectors that change sign when any two arguments are swapped. (Wewill be able to say more on that later.) An arbitrary 2-tensor can be uniquely written as thesum of a symmetric tensor and an antisymmetric tensor:

T = T s + T a ,

where

T s(v,w) =1

2

(T (v,w) + T (w,v)

),

T a(v,w) =1

2

(T (v,w)− T (w,v)

),

and T s(v,w) = T s(w,v), T a(v,w) = −T a(w,v).

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Example 7.7.

dxi ⊗ dxj =1

2

(dxi ⊗ dxj + dxj ⊗ dxi

)+

1

2

(dxi ⊗ dxj − dxj ⊗ dxi

).

The first term is symmetric, the second term is antisymmetric.

A general k-tensor with k > 2 has a more complicated decomposition into ‘fundamentalblocks’ with certain symmetry properties, but in all cases a completely antisymmetric part canbe extracted by the operation of complete alternation: T 7→ AltT where

(AltT )(v1, . . . ,vk) :=1

k!

∑σ∈Sk

(−1)σ T (vσ(1), . . . ,vσ(k)) .

Here the summation is over all k! permutations from the group Sk and (−1)σ stands for thesign of permutation. If a tensor T is already antisymmetric, the alternation will not change it:AltT = T .

Definition 7.4. The exterior product of two antisymmetric tensors T and R is defined as thealternation of their tensor product and denoted by the ∧ (read: ‘wedge’) symbol:

T ∧R := Alt (T ⊗R) .

The exterior product is often referred to as ‘wedge product’.

Theorem 7.1 (Properties of exterior multiplication). The exterior multiplication is associative:

(T ∧R) ∧ S = T ∧ (R ∧ S)

for arbitrary antisymmetric tensors T , R and S, and skew-commutative, meaning

T ∧R = (−1)pqR ∧ T

if T is a p-tensor and R is a q-tensor. It is distributive w.r.t. the sum of tensors and bilinear(over numbers, at a point, and over functions, on a manifold). The exterior products of thedifferentials dxi1 ∧ . . . ∧ dxik with i1 < . . . < ik make a basis in the space of antisymmetrick-tensors at a point. (Here a k-fold exterior product is defined using the associativity property.)

We shall not give a proof of this theorem. It is a purely algebraic statement. Instead weconsider examples.

Example 7.8. For the basis differentials,

dxi ∧ dxj =1

2

(dxi ⊗ dxj − dxj ⊗ dxi

).

Example 7.9. For arbitrary 1-forms,

α ∧ β =1

2

(αiβj − αjβi

)dxi ∧ dxj .

8


Example 7.10. Check the associativity for triple exterior products of 1-forms. Consider α ∧(β∧γ) and (α∧β)∧γ. We may also introduce a provisional notation α∧β∧γ := Alt(α⊗β⊗γ).Expanding these three objects over tensor products we can see that they coincide (exercise!).

We may give the following fundamental definition.

Definition 7.5. Antisymmetric covariant tensors (on a manifold M or at a point x ∈ M)are also called differential forms or exterior differential forms. Both adjectives are commonlyomitted. In particular, an antisymmetric k-tensor is called an (exterior differential) k-form.

From Theorem 7.1 we can find the dimensions of the spaces of k-forms at a point x ∈ M .Suppose the dimension of M is n. So we have n basis differentials dx1, . . . , dxn. Consider thek-forms for all k = 0, 1, 2, . . . at x.

The 0-forms are just numbers. The dimension of this space is 1.The 1-forms have the differentials dxi, i = 1, . . . , n , as a basis. The dimension is n. (This is

the cotangent space T ∗xM.)The 2-forms have the exterior products dsi∧dxj where i < j as a basis. Hence the dimension

is n(n− 1)/2 .For arbitrary k between 0 and n, as a basis of the space of k-forms we may take the exterior

products dxi1 ∧ . . . ∧ dxik with i1 < . . . < ik, and since there are(nk

)of such products, the

dimension is(nk

).

In particular, the space of n-forms, where n = dimM , is spanned by dx1 ∧ . . . ∧ dxn and is1-dimensional. All k-forms with k > n are zero. (We may agree to a convention that k-formswith negative k are zero as well. This allows to speak of k-forms for arbitrary integer k.)

Example 7.11. For a 1-dimensional manifold, there are only 0-forms and 1-forms that arenon-zero. The dimensions (at a point) are 1 and 1. We may write f for 0-forms and fdx for1-forms.

Example 7.12. For a 2-dimensional manifold, there are 0-forms, 1-forms and 2-forms that canbe non-zero. The dimensions are 1, 2 and 1. We may write f for 0-forms, Pdx+Qdy for 1-forms,and g dx ∧ dy for 2-forms.

Example 7.13. For a 3-dimensional manifold, there are 0-forms, 1-forms, 2-forms and 3-formsthat can be non-zero. The dimensions are 1, 3, 3 and 1. We may write f for 0-forms,

A = A1 dx+ A2 dy + A3 dz

for 1-forms,B = B1dy ∧ dz −B2 dx ∧ dz +B3 dx ∧ dy

for 2-forms, and H = g dx ∧ dy for 3-forms. (The notation for 2-forms is chosen so to be closeto the traditional notation used in vector calculus.)

Example 7.14. For a 4-dimensional manifold, there are 0-forms, 1-forms, 3-forms and 4-formsthat can be non-zero. The dimensions are 1, 4, 6, 4 and 1. Exercise: write the bases if thecoordinates are denoted t, x, y, z (mimicking special relativity).

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For denoting forms, besides capital Latin letters such as A, B, etc., the lower-case Greekletters, e.g., σ, τ , and α, are often used.

An arbitrary k-form ω is expanded over a basis as

ω =∑

i1<...<ik

ωi1...ik dxi1 ∧ . . . ∧ dxik .

There are(nk

)independent components ωi1...ik of a k-form. We have to keep the summation

symbol∑

because the summation is not over all possible combinations of indices and hence notcovered by the Einstein summation convention. For this reason it is sometimes convenient tore-write this sum allowing the indices i1, . . . , ik to run independently. Then we need to agreeabout the meaning of ωi1...ik for the indices not in the increasing order.

Convention. For components ωi1...ik defined initially only for the indices i1 < . . . < ik (in theincreasing order), we set

ωσ(i1)...σ(ik) := (−1)σωi1...ik

for all permutations σ ∈ Sk, and we define ωi1...ik := 0 if any two indices repeat. (The resultingarray ωi1...ik is defined for all possible values of indices and is antisymmetric w.r.t. to them.)

Proposition 7.2. Under the above convention,

ω =∑

i1<...<ik

ωi1...ik dxi1 ∧ . . . ∧ dxik

=1

k!ωi1...ik dx

i1 ∧ . . . ∧ dxik .

Here in the first line the summation is extended over the increasing indices only and in the secondline the summation is over all indices.

Example 7.15. Consider the 2-forms. Suppose we have

B =∑i<j

Bij dxi ∧ dxj .

We define Bij := −Bji if i > j and Bii := 0. Then we have Bij for all values of i and j, andBij = −Bji for all i, j. We can transform the expansion as follows:

B =∑i<j

Bij dxi ∧ dxj =

1

2

∑i<j

Bij (dxi ∧ dxj − dxj ∧ dxi) =

1

2

∑i<j

Bij dxi ∧ dxj − 1

2

∑i<j

Bij dxj ∧ dxi =

1

2

∑i<j

Bij dxi ∧ dxj − 1

2

∑j<i

Bji dxi ∧ dxj =

1

2

∑i<j

Bij dxi ∧ dxj + 0 +

1

2

∑i>j

(−Bji) dxi ∧ dxj =

1

2

(∑i<j

Bij dxi ∧ dxj +

∑i=j

Bij dxi ∧ dxj +

∑i>j

Bij dxi ∧ dxj

)=

1

2

∑i,j

Bij dxi ∧ dxj =

1

2Bij dx

i ∧ dxj .

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We omit a proof of Proposition 7.2. In the general case it goes exactly as in the case consideredin the example above.

Forms give an example of the following algebraic structure.

Definition 7.6. A collection of vector spaces (Vk) indexed by k ∈ Z is called a graded (orZ-graded) vector space. If A = (Ak) is a graded vector space and a bilinear operation

Ak × Al → Ak+l

is defined for all k and l, then A is called a graded algebra. A graded algebra is associative ifa(bc) = (ab)c whenever the products are defined and skew-commutative (or graded-commutative)if ab = (−1)pq ba for all a ∈ Ap and b ∈ Aq. (Note that the space A0 is automatically an algebra,in the ordinary sense.)

Differential forms (on a manifold or at a point) make a skew-commutative graded algebraw.r.t. the exterior product. The graded algebra of all forms on M is traditionally denoted Ω∗(M)or Ω(M) (without the ‘star’), though there are other notations 3. The space of the k-forms fora particular k is denoted Ωk(M). Hence we have a graded algebra Ω(M) =

(Ωk(M)

).

7.3 Pull-back of tensors and forms

Our aim is to show that a map of manifolds F : M → N induces a linear map F ∗ : Ωk(N) →Ωk(M) for each k = 0, 1, . . . , n (in the opposite direction), which preserves the exterior multipli-cation, and to describe it explicitly.

First of all, 0-forms are just functions, Ω0(M) = C∞(M). So we have their pull-back

F ∗ : Ω0(N)→ Ω0(M)

which is an algebra homomorphism.Now, consider 1-forms. Each 1-form α ∈ Ω1(N) is, in particular, a function on TN (satisfying

an extra property that it is linear on each tangent space). The map F : M → N generates thetangent map TF : TM → TN . We define the pull-back on 1-forms

F ∗(α) := (TF )∗(α) ,

where at the r.h.s. there stands the ordinary pull-back of functions.

Proposition 7.3. The pull-back maps 1-forms to 1-forms. In coordinates, the pull-back of a1-form amounts to the substitution both in the arguments of its components and in the basisdifferentials.

Proof. Consider a 1-form α on N . We have F ∗(α)(x,v) = α(TF (x,v)) = α(F (x))(dF (x)(v)) =αa(F (x))∂y

a

∂xivi, so F ∗(α)i = αa(F (x))∂y

a

∂xi, and F ∗(α) = αa(F (x))∂y

a

∂xidxi = αa(F (x)) dya(x).

3For forms at a point x ∈ M there is no standard notation. The algebra of forms at x may be denoted byΛ(T ∗

xM) and the space of all k-forms at x may be denoted Λk(T ∗xM). Here, for a vector space V , Λ(V ) is the

symbol of the so-called ‘exterior algebra’ of V and Λk(V ) is the symbol of the ‘kth exterior power’ of V . (Thereis a graded algebra Λ(V ) = (Λk(V )) for an abstract V . You may guess its construction.)

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Example 7.16. Consider a 1-form dg, where g ∈ C∞(N). For a pull-back we have: F ∗(dg)(x,v) =dg(F (x))(dF (x)(v)) = d(g F )(x)(v) = d(F ∗g)(x)(v). Hence

F ∗(dg)

= d(F ∗g

).

The same conclusion can be obtained using coordinates:

F ∗(dg) = F ∗(∂ag dy

a)

= ∂ag(F (x))∂ya

∂xidxi = d

(F ∗g

).

To see how the pull-back is defined on k-forms, consider the more general case of arbitrarycovariant k-tensors. They are multilinear functions on tangent spaces (for each point of themanifold). To treat them similarly to the 1-forms, consider the k-fold ‘product over M ’

TM ×M . . .×M TM := (x; v1, . . . ,vk) | ∀i vi ∈ TxM .

Clearly it is a manifold of dimension n+kn. A covariant k-tensor on M is nothing but a functionon this manifold with the property that it is linear w.r.t. to each vector argument. The tangentmap TF : TM → TN obviously extends to a map

T (k)F : TM ×M . . .×M TM → TN ×N . . .×N TN ,

(x; v1, . . . ,vk) 7→(F (x); dF (x)(v1), . . . , dF (x)(vk)

).

For a covariant k-tensor S on N define its pull-back by the formula

F ∗(S) := (T (k)F )∗(S) ,

where at the r.h.s. there stands the ordinary pull-back of functions. We often write F ∗S forF ∗(S), omitting the brackets.

Proposition 7.4. The pull-back of a k-tensor on N is a k-tensor on M . The pull-back mapsantisymmetric tensors to antisymmetric tensors. In coordinates, the pull-back of k-tensors (inparticular, k-forms) amounts to the substitution in the arguments of the components and in thedifferentials.

Proof. Consider k = 2 for the simplicity of notation. Consider an arbitrary smooth mapF : M → N . It induces the map

T (2)F : TM ×MTM → TN ×

NTN .

We need to check that the pull-back

(T (2)F )∗ : C∞(TN ×

NTN

)→ C∞

(TM ×

MTM

)applied to a 2-tensor S (i.e., a function bilinear in the vector arguments) gives again a 2-tensor.Indeed, we have

F ∗S(x; v1,v2) =((T (2)F )∗S

)(x; v1,v2) = S

(T (2)F (x; v1,v2)

)=

S(F (x); dF (x)(v1), dF (x)(v2)

),

12


which is clearly bilinear in v1,v2 because the tangent map dF (x) is linear and the functionS(y; w1,w2) is bilinear in w1, w2, for arbitrary w1 and w2. So tensors are indeed mappedto tensors. If a function S(y; w1,w2) is antisymmetric in w1, w2, it follows that so is thefunction F ∗(S)(x; v1,v2) with respect to v1 and v2. Hence the antisymmetric tensors (exteriordifferential forms) are mapped to the antisymmetric tensors. Finally, suppose that in coordinates,S = S(y) dya ⊗ dyb. Then we have, similarly to Proposition 7.3 ,

(F ∗S)(x; v1,v2) = S(F (x; dF (x)(v1), dF (x)(v2)

)=

Sab(F (x)

) ∂ya∂xi

vi1∂yb

∂xjvj2 = Sab

(F (x)

) ∂ya∂xi

∂yb

∂xjvi1 v

j2 ,

hence

(F ∗S(x))ij = Sab(F (x)

) ∂ya∂xi

∂yb

∂xj

or

(F ∗S(x))ij dxi ⊗ dxj = Sab(F (x)

) ∂ya∂xi

∂yb

∂xjdxi ⊗ dxj =

Sab(F (x)

) ∂ya∂xi

dxi ⊗ ∂yb

∂xjdxj = Sab

(F (x)

)dya(x)⊗ dyb(x) ,

which is exactly the substitution into the argument y of the tensor components Sab and into thedifferentials dya, whose tensor products dya ⊗ dyb make a basis in the space of all 2-tensors ateach point.

For the k-forms, i.e., the antisymmetric k-tensors, we need to make a substitution into eachdifferential in the wedge products dya1 ∧ . . .∧dyak . From Proposition 7.4 also follows that tensorproducts of covariant tensors are mapped to tensor products, and wedge products of forms, towedge products. In a more abstract language, for an arbitrary map F : M → N , the pull-backF ∗ : Ω(N)→ Ω(M) is a homomorphism of graded algebras.

Proposition 7.5. The pull-back of k-forms (actually, of any covariant k-tensors) satisfies

(G F )∗ = F ∗ G∗ .

Proof. The identity (G F )∗ = F ∗ G∗ holds for pull-backs of arbitrary functions and since thepull-back of tensors is defined simply as the restriction to them of the pull-back of functions, thisidentity holds for tensors and forms automatically. (The key statement is Proposition 7.4.)

Remark 7.6. As Example 7.16 shows, for any function g the pull-back F ∗(dg) of its differentialdg is the differential d(F ∗g) of the pull-back F ∗g. In other words, we have a commutativediagram:

C∞(M)d−−−→ Ω1(M)

F ∗

x xF ∗C∞(N)

d−−−→ Ω1(N)

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7 Di erential forms: algebraic theory - School of Mathematicstv/Teaching/Differentiable...

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