Definition 6

A law chunk of left operation of class $C^r$ of G on X is a mapping $\psi$ defined on an open subset $\Omega$ of $G \times X$ containing $\{e\} \times X$, with values in X and with the following properties: (i) $\psi$ is of class $C^r$; (ii) for all $x \in X$, $\psi(e, x) = x$; (iii) there exists a neighbourhood $\Omega_1$ of $\{e\} \times \{e\} \times X$ in $G \times G \times X$ such that, for $(g, g', x) \in \Omega_1$, the elements of $m(g, g')$, $\psi(m(g, g'), x)$, $\psi(g, \psi(g', x))$ are defined and $\psi(g, \psi(g', x)) = \psi(m(g, g'), x)$.

Law chunks of right operation of class $C^r$ are defined similarly. We often write $gx$ instead of $\psi(g, x)$. Let G' be a Lie subgroup germ of G and X' a submanifold of X. Suppose that the set $\Omega'$ of $(g, x) \in \Omega \cap (G' \times X')$ such that $\psi(g, x) \in X'$ is open in $G' \times X'$ (a condition which is always fulfilled if $X'$ is open in $X$). Then $\psi|_{\Omega'}$ is a law chunk of left operation of class $C^r$ of $G'$ on $X'$, which is said to be derived from $\psi$ by restriction to $G'$ and $X'$.