What Defines a One-Form in Multivariable Calculus and Differential Geometry?

[pellman]

A one-form is something of the form

$$\omega=\omega_\mu dx^\mu$$

But is it necessary that the components $$\omega_\mu$$ be components of a type (0,1) tensor?

For instance, the connection one-form is defined to be

$${\omega^{\alpha}}_\beta = {\Gamma^\alpha}_{\gamma\beta} \hat{\theta}^\gamma$$

where $$\hat{\theta}^\gamma$$ is a basis of the dual tangent space, though not necessarily a coordinate basis. Here the components $${\Gamma^\alpha}_{\gamma\beta}$$--the connection coefficients, i.e., Christoffel symbols-- not only are not those of a type (0,1) tensor, they are not even those of a tensor.

So is this legitimately a one-form?

 

[Hurkyl]

I think with any vector bundle E over your manifold M, it's fair to call any (bundle) mapping TM-->E a "one-form". (i.e. a function that takes tangent vectors and maps them to E-vectors)

For the usual one-forms, E is just the trivial line bundle MxR-->M -- that is, the one in which scalar fields live. Each (usual) one-form takes a tangent vector field on M and maps it to a scalar field on M. T*M is the bundle in which all such one-forms live.

 

[pellman]

Thank you, Hurkyl. I'm getting there, I really am. But I am in chap 7 of Nakahara's Geometry, Topology, and Physics and fibre bundles and such are in chap 9. I think you are saying that the answer to "is it necessary that the components $$\omega_\mu$$ be components of a type (0,1) tensor? " is "No". Is that right?

Each (usual) one-form takes a tangent vector field on M and maps it to a scalar field on M. T*M is the bundle in which all such one-forms live

Then again, maybe this means "Yes, it is necessary"? The elements of T*M are identical with the (0,1) tensors, aren't they?

 

[astros]

(...)
For instance, the connection one-form is defined to be

$${\omega^{\alpha}}_\beta = {\Gamma^\alpha}_{\gamma\beta} \hat{\theta}^\gamma$$
(...)

I heared "connection one-form" but i see 2 components "two indices" ! $${{\omega^{\alpha}}_\beta}'s$$ are not one-forms and are not expressed in the natural basis (dx) they are the components of a 1-1 tensor and a form has never been a 1-1 tensor ! I think

 

[pellman]

I think maybe it is just sloppy use of the term "one-form". Though I think it is standard. The wikipedia entry just calls it "connection form" instead of "connection one form" But it is linear in the one-form basis elements $$\hat{\theta}^\alpha$$. It is not a k-form where k>1.

On the other hand, maybe any linear combination of the $$\hat{\theta}^\alpha$$ is a one form, even though the coefficients are the not components of a (0,1) tensor. I'm not able to glean how strict the definition of a one-form is from the sources I have checked.