Tensor Fields and Lie Derivative

Posted on 2025-06-19 In Math

Outline

Section 5: Tensor Fields; Lie Derivative

5 Tensor Fields; Lie Derivative

The usual algebraic operations that one performs on vector spaces may also be done

on vector bundles. Thus if E and F are dierentiable vector bundles, then their tensor

product E ⊗ F may be dened as follows. Consider the set E ⊗ F =



x∈M

⊗ F

It can be made into a dierential manifold by noting that when E varies over a small

enough open set U around any point, this union can be identied with U ×(R

⊗R

) by

using local trivialisations of E and F . Identifying R

⊗ R

with R

, we may introduce

a dierentiable structure on



x∈U

⊗ F

. It is easy to see that these structures do

not depend on the particular local trivialisations one uses, and consequently glue up

to provide a dierential structure on E ⊗ F . It is clear how to dene a vector bundle

structure on this space. The dual E

∗

as well as the tensor, exterior and symmetric

powers of a vector bundle E may also be dened in a similar way. If E and F are the

locally free sheaves of A-modules corresponding to E and F , then the tensor product

E ⊗ F is the vector bundle associated to the presheaf which assigns to any open set U

the A(U )-module E(U) ⊗

A(U)

F(U ). Its stalk at any point x ∈ M is easily seen to be

⊗

On a dierential manifold M , many interesting geometric objects are described by

sections of vector bundles, usually the tensor powers of the tangent bundle and its dual,

the cotangent bundle. We will only deal with dierentiable sections even if we do not

say so explicitly each time.

Sections of the cotangent bundle are called dierential forms of degree 1 and sections

of its p-th exterior power Λ

∗

) are called dierential forms of degree p. Note that

there are no nonzero dierential forms of degree greater than the dimension of M. A

dierential form of degree p may also be regarded as an alternating A(M )-multilinear

form of degree p on the space of vector elds (i.e. takes the value 0, whenever two of

the argument vector elds are equal). With this interpretation, one can see that the

exterior product of two dierential forms α, β of degree p and q is given by

(α ∧ β)(X

, . . . , X

p+q

)



α(X

σ (1)

, . . . , X

σ (p )

)β(X

σ (p +1)

, . . . , X

σ (p +q)

)

where σ runs through the so-called ‘shue’ permutations and ϵ

is the signature of

σ. (A shue is a permutation which preserves the relative orders in the two subsets,

that is to say, it is a permutation of {1, . . . , k + l} such that σ(1) < ··· < σ(p) and

σ(p + 1) < ··· < σ(p + q).)

In order to develop dierential calculus on a dierential manifold M , we rst need

the notion of dierentiation of a tensor eld. We then seek to dene the derivative of a

tensor eld, namely a section of



T ⊗

∗

, with respect to a vector eld X. The most

natural denition that one might give, following the geometric denition of a vector eld

Chapter II: Differential Operators

as an innitesimal transformation given above, is the following.

5.2. Denition. If ω is a tensor eld, then the Lie derivative L(X)(ω) of ω with

respect to X is

lim

t→0

(φ

∗

ω − ω)

where φ

is the ow determined by X.

It is not dicult to show that the above limit exists if (as we assume) the vector

eld X and the tensor eld in question are dierentiable. Note that the automorphisms

give also isomorphisms φ

∗

between tensors at any point m ∈ M and those at φ

(m).

With this understanding, the above denition amounts to saying that the value of the

tensor eld L(X)(ω) at any point m ∈ M is the tensor lim

t→0

(φ

∗

(m)

−ω

)

. This

is the obvious generalisation of the action of X on functions, that is to say, we have

L(X)f = Xf for functions f .

Imitating the proof for the product rule for dierentiation, one can show the follow-

ing.

5.3. Proposition. Assume given an A-bilinear sheaf homomorphism (ω, ω

′

) 7→

B(ω, ω

′

) which associates to any two tensor elds ω, ω

′

of xed types, a third one.

Assume that for any dieomorphism φ of an open submanifold U with U

′

, we have

B(φ

∗

ω, φ

∗

′

) = φ

∗

B(ω, ω

′

). Then for any vector eld X on M , the following identity

holds:

L(X)B(ω, ω

′

) = B(L(X)ω, ω

′

) + B(ω, L(X)ω

′

Proof. In fact, by denition, (L(X)B(ω, ω

′

))

= lim

t→0

∗

B(ω, ω

′

)

(p)

− B(ω, ω

′

)

= lim

t→0

B(φ

∗

(p)

, φ

∗

′

(p)

) −B(ω

, ω

′

)

= lim

→

B(φ

∗

(p)

− ω

, φ

∗

′

(p)

) + B(ω

, φ

∗

′

(p)

− ω

′

)

= B



lim

t→0

∗

(p)

− ω

, φ

∗

′

(p)



+ B



, lim

t→0

∗

′

(p)

− ω

′



= B(L(X)ω, ω

′

)

+ B(ω, L(X)ω

′

)

5.4. Corollary.

Section 5: Tensor Fields; Lie Derivative

i) If ω, ω

′

are tensor elds, then

L(X)(ω ⊗ ω

′

) = L(X)ω ⊗ ω

′

+ ω ⊗ L(X)ω

′

ii) If α, β are dierential forms, then

L(X)(α ∧ β) = L( X)α ∧β + α ∧ L(X)β.

iii) If ω is a dierential form of degree 1 and X, Y are vector elds, then

X(ω(Y )) = (L(X)ω)(Y ) + ω(L(X)Y ).

Proof. We simply have to take in the above proposition the map B to be

i) ( ω, ω

′

) 7→ ω ⊗ ω

′

;

ii) ( α, β) 7→ α ∧β;

iii) ( ω, Y ) 7→ ω(Y ).

5.5. Remark. From the denition of Lie derivation it is easy to deduce that if

L(X)ω = 0, then ω is left invariant under the ow of X, i.e. φ

takes ω to itself. Hence

we give the following denition.

5.6. Denition. Let X be a vector eld and ω a tensor eld. Then we say that ω is

invariant under X if L(X)ω = 0.

5.7. Computation of L(X). In Proposition 5.2, we assumed that B is A-bilinear,

but used mainly that it was R-bilinear. The only point at which we used A-linearity was

when we took the limit inside the argument in B. Notice that if X, ω are dierentiable

(as we always assume), then the limit as t tends to zero, of

(φ

)

∗

ω − ω

, namely

(

)(

)

exists even uniformly on compact sets. Indeed this exists even uniformly for the partial

derivatives. We do not wish to go into this question extensively because it is irrelevant

to the present discussion. We mention it only to point out that the conclusion of

Proposition 5.2 is valid even when B is not A-bilinear but satises a suitable continuity

axiom. We will apply this to the following examples in which in any case the identity

can be directly checked.

5.8. Examples.

1) ( Y, f ) 7→ Y f where Y is a vector eld and f is a function. The corresponding

identity gives X(Y f ) = L(X)(Y ) + Y Xf , which computes the Lie derivatives on

Chapter II: Differential Operators

vector elds to be

L(X)(Y ) = XY −Y X.

2) ( Y, Z) 7→ [Y, Z], where Y, Z are vector elds. Then we get, using the above com-

putation, that

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]]

which is just the Jacobi identity.

3) The above identity can also be rewritten as

L([X, Y ]) = L(X)L(Y ) −L(Y )L(X)

on vector elds. It is easy to see that this is valid even for derivatives of other

tensor elds.

5.9. Remark. From Remark 5.4, we conclude that if X and Y are commuting vector

elds, that is to say, [X, Y ] = 0, then the ow φ

of X leaves Y xed. If ψ

is the ow of

Y , then it follows that the group φ

′

◦ψ

◦(φ

′

)

−1

gives rise to the vector eld φ(Y ) = Y .

The uniqueness of the ow therefore implies that φ

and ψ

commute for all t, t

′

Corollary 5.3 also computes L( X) on dierential forms. Firstly, on forms of degree

1, we have, by 5.4, iii),

(L(X)ω)(Y ) = Xω(Y ) −ω([X, Y ]),

while 5.4, ii) gives the computation on dierential forms of higher degree. Indeed, we

get

(L(X)α)(X

, . . . , X

) = Xα(X

, . . . , X

) −



i=1

α(X

, . . . , [X, X

], . . . , X

While the Lie derivative is a good notion of dierentiation of a tensor eld with

respect to a vector eld, it does not lead to a notion of dierentiation of a tensor eld

with respect to a tangent vector at a point. The reason for this is that the value of the

Lie derivative of a tensor eld at a point depends on the value of the vector eld not only

at that point, but in a neighbourhood. It is easy to see that from the algebraic point of

view this is due to the fact that the map X 7→ L(X) is itself a dierential operator and

Section 5: Tensor Fields; Lie Derivative

not an A-linear map. For instance, if f ∈ A(M), and α is a 1-form, we have

(L(fX)α)(Y ) = (fX)α(Y ) −α([fX, Y ])

= f (Xα(Y )) − α(f[X, Y ] −(Y f )X)

= f X(α(Y )) − fα([X, Y ]) + (Y f )α(X)

= f (L(X)α)(Y ) + (Y f)α(X).

Thus we have

L(fX)α = fL(X)α + α(X)df.

In order to get a good notion of dierentiation on tensor elds, we need some ad-

ditional structure on the manifold. We will deal with this in some detail in Chapter