Understanding the Confusion Surrounding the Definition of a k-form

[logarithmic]

Can someone explain whether, by definition, a k-form is a tensor, or a tensor field.

In "Tensor Analysis on Manifolds" (Bishop and Goldberg), it says: "A differential k-form is a $$C^\infty$$ skew-symmetric covariant tensor field of degree k (type (0,k))" [note: 0 refers to the dual space, and k refers to the original space], would suggest a k-form is a function like $$T: M \to T^0_kM$$, where M is a manifold and $$T^0_kM$$ is the set of type (0,k) tensors.

However, a 1-form df, is a map of the form $$df: T_pM \to \mathbb{R}$$ ($$T_pM$$ is the tangent space of M at p), which isn't what the above definition gives. It's consistent with the definition on wikipedia, which treats k-forms as tensors rather than tensor fields: http://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions

Adding to my confusion is some notes I have which says the space of k-forms at $$p\in M$$ is the vector space $$\Lambda^k(T_pM)$$, which is the set of functions $$T: (T_pM)^k \to \mathbb{R}$$. Yet it says that k-form fields, are called k-forms. It also then in defining the pullback uses an expression $$\omega(V_1,\dots, V_k)$$, where the $$V_i$$ are vector fields. How can you apply a vector field to a k-form, when its domain is a point on a manifold (or k-tuple of elements from a tangent space, under the other definition)? Is this just loose notation to actually mean the value of the vector field, i.e., $$V_i(p)$$ for some p in M, which would actually be an element of the tangent space, instead of $$V_i$$. In which case the definition of a k-form being a tensor, rather than a tensor field was used.

So can someone clear up all these conflicting usages and definitions of a k-form for me?

 

[arkajad]

You must be prepared that it can mean both. Being aware of that always look at the context. It is much like when we say: "let's calculate the derivative". It is from the context that you have to deduce whether it is about the derivative at a given point or the derivative as a function. Usually, when you are paying attention to the context, there is only one way in which a given reasoning makes sense.