One-parameter local group of transformations

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

ϕ : U \to M,

where $U$ is an open subset of $R \times M$ of the form

U = ⋃_{x \in M} (I (x) \times {x}),

where $I (x) = (ϵ_{-} (x), ϵ_{+} (x))$ is an open interval containing 0:

ϵ_{-} (x) < 0 < ϵ_{+} (x) .

The mapping satisfies:

For all $x \in M$ ,

ϕ (0, x) = x,

For all $s, t$ (where both sides are defined),

ϕ (s, ϕ (t, x)) = ϕ (s + t, x),

For all $t$ (where both sides are defined),

ϕ (- t, x) = ϕ (t, \cdot)^{- 1} (x),

that is, $ϕ (- t, x)$ is the inverse transformation of $ϕ (t, x)$ at $x$ .

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 $D i f f (M)$ .