Consider a smooth manifolds \(M\). Suppose \(M\) is a smooth \(n\)-manifold. A function \(f\colon M\to \mathbb{R}^k\) is a smooth function if for every \(p\in M\), there exists a smooth chart \((U,\varphi)\) for \(M\) with \(p\in U\) such that \(f\circ \varphi^{-1}\) is smooth on \(\varphi(U)\). The function \(f\circ \varphi^{-1}\colon \varphi(U)\to \mathbb{R}^k\) is called the coordinate representation of \(f\).

Remarks
  • The constant function is smooth.
Remarks
  • In the same manner one can define \(C^k\) or analytic functions provided a regular enough manifold.
  • It follows from the smoothness between transition maps, that every coordinate representation is smooth. Therefore, the regularity of a function is a geometric invariant.
  • The class of all smooth functions on \(M\) is denoted by \(C^\infty (M)\).

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