Theorem 1.2 (Whitney). All n-dimensional manifolds are surfaces in R for suitably largeN ≥ n, N ≤ 2n+1. Definition 1.3. Let M,M̃ be manifolds of dimension n and ñ respectively. A map f ∶M → M̃ is said to be smooth if φ̃β ○ f ○ φ α ∶ R → R is smooth for all α,β. Definition 1.4. Let γ ∶ R → M be a smooth curve with γ(0) = p ∈ M . The tangent vector V to γ at p is defined by dγ( ) d ∣ =0 = V ∣ p ∈ TpM.
