Submanifold

Let $𝑁$ be a $𝑛$ -dimensional $𝐶 𝛼$ differentiable manifold and let $1 \leq 𝑚 \leq 𝑛$ . A $𝑚$ -dimensional (embedded) $𝐶 𝛼$ submanifold $𝑀 \leq 𝑁$ is a subset such that every $𝑝 \in 𝑀$ has a coördinate chart $(𝑥, 𝑈)$ such that $𝑥 (𝑈 \cap 𝑀) = 𝑥 (𝑈) \cap (ℝ 𝑚 \times {⃗ 𝟎})$ . diff

Thus $𝑀$ inherits the structure of an $𝑚$ -dimensional $𝐶 𝛼$ differentiable manifold from $𝑁$ , with the induced chart $(¯ 𝑥, 𝑈 \cap 𝑀)$ .

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Submanifold

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