when limited to the tangent space of a certain point, differential forms are multi-linear functions. While the existence of a global smooth form depends on the topology of the underlying manifold.
I'm teaching myself from Spivak's Calculus on Manifolds, and am having trouble understanding differential forms the way he's explaining them. Are they multilinear functions still? Or are they something completely different? Or are they a formalization of vector fields?
Hi Enderwiggin!
I suggest you read Marian Fecko's 'Differential Geometry and Lie Groups for Physicists'. This is a great book!!! You will clearly know almost everything in Differential Geometry if you read it and do the exercises in the book!!!
1.If you are dealing with the vector spaces defined on a single point of a manifold, the covectors are just the elements in the dual space of the tangent vector space. This tanget vector space is actually a finite dim linear algebra over R. Further more, you are able to use the multi-linear map to define tensors on that point. These tensors span a linear space over R. Using the tensor product, tensors form the Tensor Algebra over R.Then, use the anti-symmetric operator(this is actually a projection operator) act on the tensor of type (0,p). You get some anti-symmetric 'covectors'. These anti-symmetic (0,p) tensors with the wedge product form an algebra, which is called the exteria algebra.Those elements are called the forms on a single point...
2.If you are dealing with the vector field, then you will see that the vector fields an covector fields are actually infinite dim algebras over R. We cannot use a infinite dim stuff. So, as we already know that the functions defined on a chart of a manifold forms a infinite dim linear algebra, called F(M), we then just treat the (co)vector fields as the Modules over F(M) with finite generators! Then you are able to easily use some F-linear maps to define the tensor fields... and use anti-symmetric operators to get he form fileds!
Hi Enderwiggin!
I suggest you read Marian Fecko's 'Differential Geometry and Lie Groups for Physicists'. This is a great book!!! You will clearly know almost everything in Differential Geometry if you read it and do the exercises in the book!!!
1.If you are dealing with the vector spaces defined on a single point of a manifold, the covectors are just the elements in the dual space of the tangent vector space. This tanget vector space is actually a finite dim linear algebra over R. Further more, you are able to use the multi-linear map to define tensors on that point. These tensors span a linear space over R. Using the tensor product, tensors form the Tensor Algebra over R.Then, use the anti-symmetric operator(this is actually a projection operator) act on the tensor of type (0,p). You get some anti-symmetric 'covectors'. These anti-symmetic (0,p) tensors with the wedge product form an algebra, which is called the exteria algebra.Those elements are called the forms on a single point...
2.If you are dealing with the vector field, then you will see that the vector fields an covector fields are actually infinite dim algebras over R. We cannot use a infinite dim stuff. So, as we already know that the functions defined on a chart of a manifold forms a infinite dim linear algebra, called F(M), we then just treat the (co)vector fields as the Modules over F(M) with finite generators! Then you are able to easily use some F-linear maps to define the tensor fields... and use anti-symmetric operators to get he form fileds!
Remember that this time the tnsor fields are all modules over F(M)! Then, use the wedge and direct sum, you get the algebra of forms: Cartan Algebra.
Futher more, if you have Cartan Algebra, you can define the exteria derivative operator. Then you get the de-Rahm cohomology group and Poincare Lemma. These two stuff are related to the topology of the manifold.