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logarithmic
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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?
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?