Problem of the Week #177 - October 20, 2015

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Here is this week's POTW:

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Let $(M,g)$ be a Riemannian manifold. A vector field $X$ on $M$ is a Killing field if the Lie derivative of $g$ along $X$ is zero, i.e., $\mathcal{L}_Xg = 0$. Show that the Lie bracket of two Killing fields on $M$ is a Killing field.

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Honorable mention goes to Kokuhaku for recognizing an Lie derivative identity which trivializes the problem. You can read my solution below.

Spoiler

Let $X$ and $Y$ be vector fields on $M$. Then

$$\mathcal{L}_{[X,Y]}g = [\mathcal{L}_Xg,\mathcal{L}_Yg].\tag{*}$$

This identity can be obtained by using Cartan's formula $\mathcal{L}_Z = i_Zd + di_Z$ or using local coordinate expression

$$\mathcal{L}_Z(g_{ij}dx^i\wedge dx^j) = (Zg_{ij})dx^i\wedge dx^j + g_{ij}\, d(Zx^i)\wedge dx^j + g_{ij}\, dx^i\wedge d(Zy^j).$$

With identity $(*)$ at hand, the result follows since if $X$ and $Y$ are Killing, then $$\mathcal{L}_{[X,Y]}g = [\mathcal{L}_Xg,\mathcal{L}_Yg] = [0,0] = 0.$$

and thus $[X,Y]$ is Killing.