We have a number of other constructions similar to the tangent bundle that will come in handy. These are all sort of analogues of certain constructions we already know about on vector spaces. Let’s review these first.
where we have copies of the vector space and copies of the dual space . Vectors in , then, are tensors of type , while vectors in the dual space are tensors of type .
We also know about the space of antisymmetric tensors of rank over a vector space. In particular, we’re most interested in carrying this construction out over the dual space: . And of course we can take the direct sum of these spaces over all to get the exterior algebra .
Now, we will take these constructions and apply them to the tangent spaces to a manifold. We define the bundle of tensors of type over :
the “exterior -bundle” over :
and the exterior algebra bundle over :
The trick here is that for each of these constructions, if we have a basis of we automatically get a basis of each space , , and . If we start with a coordinate patch on , we get a basis of for each . Then, just as we did with the tangent bundle and the cotangent bundle we can come up with a coordinate patch “induced by ” on each of our new bundles. In this way, we can turn them from disjoint unions of vector spaces into manifolds of their own right, each with a smooth projection down to .
Now we can define a “tensor field of type ” on an open region as a section of the projection . That is, it’s a smooth map such that . Similarly, we define a “differential -form” over to be a section of the projection .