1-form

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold. A 1-form or covector field is something which eats vector fields and spits out scalar fields in a linear fashion. #m/def/geo/diff In more precise terms, a 1-form is a $𝐶 𝛼 (𝑀)$ -module homomorphism

𝔛 (𝑀) \to 𝐶 𝛼 (𝑀) .

As a section

The above definition is equivalent to a $𝐶 𝛼$ -section of the cotangent bundle. As a special case of Differential form, we denote the set of 1-forms on $𝑀$ as

Ω 1 (𝑀) : = Γ (𝑇 * 𝑀) .

At a point, each 1-form restricts to an $ℝ$ -linear form on the tangent space, i.e. a cotangent vector.

Proof of equivalence

proof

develop | en | SemBr

1-form

As a section

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