Smooth map

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Definition

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

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

Via commutative diagrams

A map is smooth if the following diagram commutes:

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

Where:

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

See also

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