Lie derivative $\mathcal{L}_X\alpha=0.$

Question

If $X$ is a vector field and $\alpha$ is a one-form, are there special one-forms such that $$\mathcal{L}_X\alpha=0?$$

Similarly, are there special vector fields such that $$\mathcal{L}_X\alpha=0?$$

I've read bits and pieces about this and I've come across this equation above implying some type of symmetry. Can anyone elaborate on this? Thank you!

Answer 1

If $\alpha$ is preserved by the flow $\phi_t$ of $X$, that is $\phi_t^*\alpha=\alpha$, then $L_X\alpha={d\over{dt}}\phi_t^*\alpha={d\over{dt}}\alpha=0$.

Answer 2

I want to just add on to Tsemo's answer.

If $g$ is a Riemannian metric, and $L_X g=0$ then $X$ is called a Killing vector field.

Lie derivatives are derivatives of objects with respect to a family of diffeomorphisms $\phi_t$. I've heard Lie derivatives described (by some in physics) as derivatives under infinitesimal coordinate changes.