A vector $X$ on a Riemannian manifold $(M,g)$ is called conformal if $L_{X}(g)=2sg$ where $L_{x}$ is the Lie derivative and $s$ is a real-valued function on $M$. If $s=0$, $X$ is called a killing vector field. I am not a student of differential geometry but I really want to know physical application of such vectors.
I have just get an answer link on wiki says that these fields has flow preserves the conformal structure of the manifold. So the question becomes two!!! first one what does flow preserves ... mean? I know mapping between two manifolds preserves angles but flow I do not. Second one is there is a physical such flow?
The flow of $X$ is a set $f_{t}$ of conformal transformation that preserves angles on $M$. These transformations map $M$ to $M$ as follows: Let $C_{X(p)}(t)$ be the curve that fits $X(p)$ and define $f_{t}(p)$ as a map that takes $p$ to $C_{X(p)}(t)$.
If this is the idea the first question is now clear. This vector field has a good property that it generates a conformal flew that preserves some geometric properties of $M$.
The second question: : I wand an example of such conformal flow in physics i.e. fluid or electrical flow example that is conformal on a manifold preferably non-Euclidean one.
The comment of @StevenGubkin is fine at least now.
Thanks all.