Flow of a Metric & Conformality

Question

What is the flow of a metric mathematically?

I want to be able to understand what it means to say that a metric preserves a conformal structure through the actual definition $\Theta_t^*g = \sigma(t,x)g$ (example 4.6.16) so that I can motivate the necessity of solving the conformal Killing equation $$\tfrac{d}{dt}|_{t=0}\Theta_t^*g = \mathcal{L}g = (\tfrac{d}{dt}|_{t=0}\sigma(t,x))g = \Gamma(x)g$$

I only know what to do with flows for a vector field, e.g. given $(x,y)$ the flow of rotations is

$$\theta : [0,2\pi] \times M \rightarrow M | (t,x,y) \mapsto \theta(t,x,y) = \theta_t(x,y) = (x \cos(t) - y \sin(t),x \sin(t) + y \cos(t))$$

I understand it in a hand-wavey way but I have no idea what $\Theta_t^*g $ even means mathematically, e.g. how would I work it out for a polar coordinate metric!?

Answer 1

That is not the "flow of the metric." I'm fairly certain that such a concept doesn't make sense.

$\theta_t^*\mathrm{g}$ is simply pulling $\mathrm{g}$ back along the flow of a vector field.

As you probably know, $\theta_t^*\mathrm{g}(X_p,Y_p)=\mathrm{g}((\theta_t)_*(X_p),(\theta_t)_*(Y_p))$. You can think of it like a doctor consulting another doctor on two symptoms of a disease. At point/hospital $p$, they don't know what the symptoms might mean, so they push them over to the point/hospital $\theta_t(p)$, and have them interpret it. Here, we have two tangent vectors at $p$ and we want them elsewhere, so we push them to $\theta_t(p)$ and look at them there.

Saying that $\theta_t^*\mathrm{g}=\lambda\mathrm{g}$ simply says that, by pushing forward from $p$ to $\theta_t(p)$, all you have done is rescaled everything. It's like transforming New York into Lilliput or vice versa. All the angles are the same, but everything has been made bigger or smaller. A more mathematical idea to help think about this is to remember that similar triangles in Euclidean space have matching angles, but are not necessarily the same size. They are, however, able to be transformed into each other via isometries and a rescaling.

As for explaining the conformal Killing equation, let's look at the limit. Let $\xi\in\Gamma(TM)$ be a smooth vector field that generates a local flow $\theta$. Then, $$\mathcal{L}_\xi\mathrm{g}=\lim_{\delta\rightarrow 0}\frac{1}{\delta}(\theta_\delta^*\mathrm{g}-\mathrm{g})=\lambda\mathrm{g},$$ which just means that $\xi$ preserves $\mathrm{g}$ up to rescaling by the function $\lambda$.