Given a manifold $M$, a one-parameter local group of transformations is a mapping
$$ \phi :U \to M, $$being $U$ an open set of $\mathbb{R}\times M$ of the form
$$ U=\bigcup_{x \in M } \{x\}\times \left( \epsilon_{-}(x),\epsilon_{+}(x) \right), $$with $\epsilon _ {+} ( x)>$, $\epsilon_{-} ( x)<0$ for $x \in M$, such that
$$ \phi_s(\phi_r(p))=\phi_{s+r}(q) $$ $$ \phi _ {-t} x =\phi _ {t} ^ {-1} x, $$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.
When $U=\mathbb R \times M$ 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 $Diff(M)$.
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Author of the notes: Antonio J. Pan-Collantes
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