Tangent bundle

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

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

The construction of topological and $𝐶 𝛼$ -structure is a little more involved. Let $𝒜$ be the maximal atlas for $𝑀$ , so each $(𝑥, 𝑈) \in 𝒜$ gives a $𝐶 𝛼$ -isomorphism

𝑥 : 𝑈 \to ℝ 𝑛

which induces a bijection

˜ 𝑥 : 𝜋 - 1 𝑈 \to ℝ 𝑛 \times ℝ 𝑛 (𝑝, 𝑣 𝜇 𝜕 𝜇) \mapsto (𝑥 (𝑝), (𝑣 𝜇))

which induce an atlas on $𝑇 𝑀$ . Thus $𝐴 \subseteq 𝑇 𝑀$ is open iff $˜ 𝑥 (𝐴 \cap 𝜋 - 1 𝑈)$ is open for every $(𝑥, 𝑈) \in 𝒜$ .

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Tangent bundle

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