Differentiable manifold

A $𝛼$ -differentiable manifold or $𝐶 𝛼$ -manifold $(𝑀, 𝒜)$ is a topological manifold equipped with a maximal $𝐶 𝛼$ -atlas.¹

note

If that’s confusing, the canonical example is Real coördinate space $ℝ 𝑛$

Embedded manifolds

By the Whitney embedding theorem, we may alternatively consider $𝐶 \infty$ -manifolds as embedded into real coördinate space, i.e. as a Real embedded manifold.

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