Differential forms on manifolds
We have been thinking of differential forms on . Hubbard defined them as follows:
Definition. A -form on is a function that takes vectors in and returns a number, such that is multilinear and antisymmetric.
I want to extend this notion to work on manifolds.
Let’s use less patronising language then, and call a -covector of a vector space , a -linear map satisfying
for the symmetric group. (Make sure you know what -linear means)
- Check that the space of -covectors of , denoted , is a vector space.
- If is a basis for , make sure you know why
is a basis for , where is the 1-covector that sends to 1, and to 0 for all . Remember, the wedge symbol , is just formal notation to remind you that the covector is antisymmetric.
Observe that setting in the definition of a -covector returns precisely the definition Hubbard gave of a -form on . The idea now is to extend this to general manifolds which we do by considering the tanget space.
Definition. A -form on a manifold is a function that assigns to each point a -covector in
So at each point on our manifold, we have a map that takes vectors in the tangent space to . This is what I mean in class when I talk about sections of the cotangent bundle.
Remark. In mathematics we have the notion of a dual. Given a vector space with basis , we have its dual vector space, denoted, with basis where, as above, (the Kronecker delta function). It is common to use the prefix “co” to talk about the dual concept, for instance, is a vector, is a covector. In the same way we have the tangent space and the cotangent space, denoted . To that end, let’s bring our notation above in line with the prevailing convention and denote
(Exercise. What should the relationship be with ?)
Example. Okay – well is a manifold (someone even claimed it was their favourite manifold!), so we better make sure the two notions we have agree. Luckily the situation here is rather simple as the tangent space at a point in is isomorphic to , so there is nothing to do. We can do better and give a description of -forms on that is more in line with general manifolds. At each point we have a standard basis for the tangent space ,
Let be the dual basis, that is, . Then we have a basis for the vector space
And so a -form on , is a function sending of the form,
, where is a function of
So for example, is a 2-form on , which takes to
The situation on a general manifold is not much harder, since we have a system of local coordinates for . The standard way to do this is to consider a system of charts on your manifold, giving you your local coordinates, but I’ll save that for another post as this one is already getting a bit long. The upshot is that for a manifold we can define a -form as a function taking points to -covectors .