One Parameter Group of Diffeomorphism

One Parameter Group of Diffeomorphism

A 1-parameter group of diffeomorphisms of a manifold $M$ is a $C^{\infty }$ map such that

• for each $t\in \mathbb{R}$ the map ${\phi }_t:M\to M$ defined by ${\phi }_t(q)=\phi (t,q)$ is a diffeomorphism of $M$ onto itself,
• for all $t,s\in \mathbb{R}$ we have ${\phi }_{t+s}={\phi }_t\circ {\phi }_s$. Given a 1-parameter group $\phi$ of diffeomorphisms of $M$ we define a vector field $X$ on $M$ as follows. For every smooth real valued function $f$ let $X_q(f)={\lim }_{h\to 0}\frac{f({\phi }_h(q))-f(q)}{h}.$ This vector field $X$ is said to generate the group $\phi$.

Note that this definition is actually just the flow generated by vector field.

Lemma

A smooth vector field on $M$ which vanishes outside of a compact set $K\subset M$ generates a unique 1-paramter group of diffeomorphisms of $M$.