One-parameter local group of transformations

Given a manifold $M$ , a one-parameter local group of transformations is a mapping

ϕ : U \to M,

being $U$ an open set of $R \times M$ of the form

U = ⋃_{x \in M} {x} \times (ϵ_{-} (x), ϵ_{+} (x)),

with $ϵ_{+} (x) >$ , $ϵ_{-} (x) < 0$ for $x \in M$ , such that

ϕ_{s} (ϕ_{r} (p)) = ϕ_{s + r} (q)

ϕ_{- t} x = ϕ_{t}^{- 1} x,

for all values of $s, t, r$ such that both sides of the equations are defined.

There is a one to one relation between one-parameter local group of transformations and vector fields. See flow theorem for vector fields for one of the implications. See @lee2013smooth chapter 12 for the whole approach.

Global one-parameter group of transformations

When $U = R \times M$ then it is simply called a one-parameter group of transformations, and the corresponding vector field is called a complete vector field. It can be thought as a one-parameter subgroup of $D i f f (M)$ .