Differential geometry MOC

Real embedded manifold

In these notes, a real embedded manifold typically refers to a 𝐶∞ submanifold of real coördinate space. diff The Whitney embedding theorem provides a sense in which every real differentiable manifold may be regarded as a real submanifold.

A subset 𝑋 ⊆ℝ𝑁 is an 𝑛-dimensional real submanifold iff has charts that are 𝐶∞ diffeomorphisms as subsets of real coördinate space, i.e. for every 𝑥 ∈𝑋 there exists a neighbourhood 𝑈′ of 𝑥 in ℝ𝑁 and an open set 𝑉 ⊆ℝ𝑛 such that there exists a 𝐶∞ map 𝜑 :𝑈′ ↠𝑉 with a 𝐶∞ right-inverse 𝜓 =(𝜑 ↾𝑈′ ∩𝑋)−1.

Properties

• Total derivative
• Tangent space

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