Differentiable flow - Knowlpedia

Let $N$ be a smooth manifold. A differentiable flow on $N$ is a map $\varphi:\mathbb R\times N\to N$ that is continuously differentiable in local manifold charts. Write $\varphi_t(x)=\varphi(t,x)$ . The required flow laws are

\varphi_0=\operatorname{id}_N,\qquad \varphi_{t+s}=\varphi_t\circ\varphi_s\quad(s,t\in\mathbb R).

Time maps and local flows

Consequently each time map is invertible, with continuously differentiable inverse $\varphi_{-t}$ .

This definition is global in time: all real $t$ are allowed. A local flow only has time maps on suitable open subsets of $\mathbb R\times N$ .

Reference

Ko Honda, Differential Geometry Course Notes, §30.