Smooth map

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Definition

A map [ilmath]f:M\rightarrow N[/ilmath] between two smooth manifolds [ilmath](M,\mathcal{A})[/ilmath] and [ilmath](N,\mathcal{B})[/ilmath] (of not necessarily the same dimension) is said to be smooth if:

• [math]\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}[/math] such that [math]F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)][/math] is smooth

Via commutative diagrams

A map is smooth if the following diagram commutes:

[math]\begin{CD} M @> F > > N\\ @V \varphi V V @V V\psi V\\ \varphi(U) @> G >=\psi\circ F\circ\varphi^{-1} > \psi(V) \end{CD}[/math]

Where:

• [ilmath]G[/ilmath] is smooth
  • (given by [ilmath]G=\psi\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)[/ilmath])
• [ilmath]M,N[/ilmath] are smooth manifolds (with smooth structures) [ilmath]\mathcal{A},\mathcal{B} [/ilmath] respectively
• [ilmath](U,\varphi)\in\mathcal{A} [/ilmath]
• [ilmath](V,\psi)\in\mathcal{B} [/ilmath]

See also

• Differential of a smooth map
• Smooth function
• Smooth manifold