# What is the importance of conformal vector fields on Riemannian manifolds?

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?

• perhaps ieeexplore.ieee.org/xpl/… is relevant to you? Dec 20, 2013 at 22:17
• Flow is just a family of mappings, one for each $t$ in some interval. So it preserves something when all of those mappings do. I do not understand your second question. What do you mean by physical? I'll give you one example, perhaps it will help. In complex plane any holomorphic map is conformal. So a conformal flow here is just a smooth family of holomorphic maps, e.g. $z + te^z$. Dec 20, 2013 at 23:03
• If I got it, I think the flow of $X$ is a set $f_{t}$ of conformal transformation that preserves angles on $M$. This transformation maps a point $C_{X}(0)$ to $C_{X}(t)$. If this is the idea the first question is now clear. 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. Dec 21, 2013 at 10:32
• @StevenGubkin I have just got the article I think it is good. Thank you Dec 21, 2013 at 13:10

## 1 Answer

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.