Path: blob/main/14Manifold_Differential_Forms.ipynb
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14. Differential -forms
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
Warning: in this notebook there are many repetitions with respect to notebook 10. Although mathematically alternating forms on modules from notebook 10 are generalizations of differential forms, the SageMath code in the present notebook differs significantly from that in notebook 10.
Let be a smooth manifold.
A differential -form or simply -form on a manifold is a smooth map that, to each point , assigns -linear antisymmetric form on the tangent space .
Reminder. -linear antisymmetric forms on arbitrary modules were presented in notebook 10, and smoothness of covariant tensor fields in notebook 13.
One can use an equivalent definition:
A differential form of degree , or differential -form, on a manifold is a smooth tensor field of type on , with the antisymmetry property:
ParseError: KaTeX parse error: Undefined control sequence: \label at position 122: …. . . , X_k ), \̲l̲a̲b̲e̲l̲{}\tag{14.1} \e…for , .
Note that from (14.1) it follows that if then .
The relation (14.1) can be reformulated as
for (permutations and their signs were defined in notebook 10).
A 0-form is a smooth real-valued function on .
The set of the k-forms on a manifold , is denoted by . It is a submodule of the module of covariant tensor fields on the manifold .
Example 14.1
Let us define a differential 2-form on a 3-dimensional manifold.
The obtained module can be defined independently:
Antisymmetrization operation Alt
If is a covariant tensor field from , then we can define a differential -form called Alt() in the following way
ParseError: KaTeX parse error: Undefined control sequence: \label at position 144: …_{\sigma(k)}), \̲l̲a̲b̲e̲l̲{}\tag{14.2} \e…for .
Alt is a linear map on the module (over ) into .
To prove that for we have note that for arbitrary permutation
We can check also that if , then Alt()=.
In fact, assume that . Then
The last two observations lead to the relation
Wedge product
If is a -differential form and is an -differential form on , the exterior, or wedge, product , is defined by ParseError: KaTeX parse error: Undefined control sequence: \label at position 87: …a\otimes\eta). \̲l̲a̲b̲e̲l̲{}\tag{14.3} \e…
Remark. Some authors use different coefficients in this formula. Our version is in accordance with that in SageMath Manifolds.
Example 14.2
If and are 1-forms, applying (14.3), (14.2) and using the fact that there are only two permutations of two elements with opposite signs we have
for Consequently
Basic algebraic properties of wedge product
From the multilinearity of the tensor product and linearity of antisymmetrization it follows.
For and
Associativity of the wedge product
To check the associativity of the wedge product we need some properties of the antisymmetrization operation.
First observe that if and , then
To prove this, let us note that
First define the subgroup of ,
and compute the sum from the right hand side of the previous equality restricted to this subgroup.
so the entire sum vanishes if the sum in brackets vanishes, i.e. Alt() vanishes.
Now for fixed let us sum over the coset }.
Thus if vanishes then the sums over all cosets vanish. Since the group is a sum of cosets , the implication (Alt1) holds true.
The second ingredient of our associativity proof is the following
In fact, the linearity of Alt and the relation implies
Applying the implication (Alt1) to we obtain
Since we have checked that the value of Alt on the expression in brackets vanishes, then the implication (Alt1) gives us
i.e., the second equality from (Alt2) holds true
The first equality in (Alt2) can be proved analogously.
From (Alt2) it follows for
Analogously for . We have proved associativity of the wedge product:
for , and more generally
ParseError: KaTeX parse error: Undefined control sequence: \label at position 136: …s \alpha_r ), \̲l̲a̲b̲e̲l̲{}\tag{14.6} \e…for
For example if are local coordinates on , then since are 1-forms, then are -forms on .
Differential forms in local coordinates
Let be a local coordinate system on . According to (13.7) a -differential form possesses the local representation
ParseError: KaTeX parse error: Undefined control sequence: \label at position 66: … · ⊗ dx^{i_k}, \̲l̲a̲b̲e̲l̲{}\tag{14.7} \e…where
ParseError: KaTeX parse error: Undefined control sequence: \label at position 122: …x^{i_k}}\Big). \̲l̲a̲b̲e̲l̲{}\tag{14.8} \e…Since is antisymmetric, is also antisymmetric.
Since Alt is linear and Alt()= we have
From (14.6) it follows that (note that are 1-forms) Thus we have ParseError: KaTeX parse error: Undefined control sequence: \label at position 228: …. \end{matrix} \̲l̲a̲b̲e̲l̲{}\tag{14.9} \e…
Note that every element of the sum with increasing indices has counterparts in the sum with arbitrarily ordered indices. For arbitrary permutation (no summation here)
Elements of the set
are linearly independent, since if
then taking the basis and an increasing index sequence , we obtain
Since and are both increasing, there is only one choice of which gives a nonzero value, namely Since
we have
Anticommutativity
If and , then
This is consequence of (we use the unordered version of (14.9))
Abbreviated notations for -forms
For arbitrary differential forms in sometimes we would not want to actually write out all of the elements of the local frames of , so instead we write ParseError: KaTeX parse error: Undefined control sequence: \label at position 43: …m_I a_I dx^I .
\̲l̲a̲b̲e̲l̲{}\tag{14.10}
\… Here the stands for the sequence of increasing indices : . That is, we sum over
For example, for and we have
If and are disjoint, then we have where , but is reordered to be in increasing order. Elements with repeated indices are dropped. Using this notation we can compute the wedge product as follows ParseError: KaTeX parse error: Undefined control sequence: \label at position 152: …Ib_J\Big)dx^K. \̲l̲a̲b̲e̲l̲{}\tag{14.11} \…
Example 14.3
Let us define a differential 2-form on a 3-dimensional manifold.
The arguments of components can be omitted
The matrix of 2-form must be antisymmetric.
We can check the formula .
Example 14.4
Let us show how the antisymmetrization operation works for tensor fields .
Now we apply the antisymmetrization operation, the result is 2-form at.
The matrix of components of is a full 3-3 matrix:
and the antisymmetrized tensor has the antisymmetric matrix of components:
Example 14.5
Consider a tensor field of type on a 4-dimensional manifold and its antisymmetrization.
Let be the frame:
and let's show how the definition of the antisymmetrization works if we take in (14.2):
As we can see the sum contains 3!=6 elements with signs equal to permutations signs.
For comparison one can use the antisymmetrize
method:
Example 14.6
Define two 1-forms with symbolic components and compute their wedge product.
Wedge product :
Compare with the result obtained from definition (14.3):
Define the matrix containing the components of 1-forms in rows:
The components of the wedge product of 1-forms are minors of the matrix of components of these 1-forms.
Example 14.7
Consider the wedge product of three 1-forms (in 3-dimensional manifold).
We can recognize in the result the Laplace expansion of the determinant det so we have
det
We can check the observation using SageMath:
The unique component of the wedge product must be equal to .
Let us check it with SageMath Manifolds.
Example 14.8
Consider two 2-forms on a 4-dimensional manifold.
We define first the names of functions,
and the upper triangles of component matrices.
Recall that according to the general rule, to obtain the wedge product we use all possible strictly increasing and disjoint sequences and define where is reordered disjoint union . We have
To be less formal, to compute the wedge product we
a) take all possible components of the first 2-form with increasing permutations of two indices from (0,1,2,3);
b) multiply them by components of the second 2-form with increasing permutations of the remaining 2 indices;
c) the products are taken with + or - depending on the sign of the joined permutation of (0,1,2,3).
What's next?
Take a look at the notebook Pullback of tensor fields.