Flow on a manifold

Let $𝑀$ be a $𝐶 𝛼$ -manifold with group of diffeomorphisms $A u t 𝛼 (𝑀)$ . A (local) 1-parameter group $𝜑 ? : ℝ \to A u t 𝛼 (𝑀)$ is called a (local) flow on $𝑀$ . #m/def/geo/diff The orbit of a point $𝑝 \in 𝑀$ defines a $𝐶 𝛼$ -curve

𝛾 = 𝜑 ? (𝑝) : (𝜖 -, 𝜖 +) \to 𝑀

with $𝛾 (0) = 𝑝$ . The map

𝑝 \mapsto ˙ 𝛾 (0) = d d 𝑡 𝜑 𝑡 (𝑝) ∣ 𝑡 = 0

defines a $𝐶 𝛼$ -vector field $𝑣 𝑎 \in 𝔛 (𝑀)$ , called the infinitesimal generator of $𝜑 ?$ .

Conversely, given a vector field $𝑣 𝑎 \in 𝔛 (𝑀)$ , one can (usually) find a corresponding local flow whose orbits are called integral curves.

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