# The Unapologetic Mathematician

## One-Parameter Groups

Let $\Phi:\mathbb{R}\times M\to M$ be a differentiable map. For each $t\in\mathbb{R}$ we can define the differentiable map $\Phi_t:M\to M$ by $\Phi_t(p)=\Phi(t,p)$. We call the collection $\left\{\Phi_t\right\}_{t\in\mathbb{R}}$ a “one-parameter group” of diffeomorphisms — since it has, obviously, a single parameter — so long as it satisfies the two conditions $\Phi_0=1_M$ and $\Phi_{t_1+t_2}=\Phi_{t_1}+\Phi_{t_2}$. That is, $t\mapsto\Phi_t$ is a homomorphism from the additive group of real numbers to the diffeomorphism group of $M$. Indeed, each $\Phi_t$ is a diffeomorphism of $M$ — a differentiable isomorphism of the manifold to itself — with inverse $\Phi_{-t}$

If we define a vector field $X$ by $X(p)=\Phi_*\frac{\partial}{\partial t}(0,p)$ then $\Phi$ is a flow for this vector field. Indeed, it’s a maximal flow, since it’s defined for all time at each point.

Conversely, if $\Phi:W\to M$ is the maximal flow of a vector field $X\in\mathfrak{X}M$, then $\Phi$ defines something like a one-parameter subgroup. Indeed, “flowing forward” by $t_1$ and the again by $t_2$ is the same as flowing forward by $t_1+t_2$ along each integral curve, and so $\Phi_{t_1+t_2}=\Phi_{t_1}+\Phi_{t_2}$ wherever both sides of this equation are well-defined. But they might not, since even if both $(t_1,p)$ and $(t_2,p)$ are in $W$ for all $p\in M$ the point $(t_1+t_2,p)$ might not be. But if every integral curve can be extended for all times, then we call the vector field “complete” and conclude that its maximal flow is a one-parameter group of diffeomorphisms.