Cotangent bundle

Let $𝑀$ be a $𝐶 𝛼$ -manifold. The cotangent bundle is a vector bundle of all cotangent spaces of $𝑀$ , diff so as a set

𝑇 * 𝑀 = ∐ 𝑝 \in 𝑀 𝑇 * 𝑥 𝑀 = ⋃ 𝑝 \in 𝑀 {𝑝} \times 𝑇 * 𝑝 𝑀 .

with the $𝐶 𝛼$ -structure of the dual vector bundle of the tangent bundle.

Sheaf theory

There is an alternative Sheaf-theoretic construction in terms of the diagonal morphism.

A smooth section $𝜔 \in Γ 𝑇 * 𝑀 = Ω 1 𝑀$ of the tangent bundle is called a 1-form. A general Differential form is a smooth section of the Exterior algebra bundle $⋀ ∙ 𝑇 * 𝑀$ .