= Kaluza-Klein decomposition along a Killing field
{c}
{title2=$h=V(dx+A)^2+\gamma$}
Where a <Killing vector field> is $\partial_x$ and its squared norm $V=h_{xx}$ is nonzero, an invariant <metric tensor> decomposes as $h=V(dx+A)^2+\gamma$. The quotient one-form has $A_i=h_{xi}/V$ and the quotient metric has $\gamma_{ij}=h_{ij}-h_{xi}h_{xj}/V$. All coefficients are independent of $x$. Changing the fibre coordinate to $x'=x+\lambda$ sends $A$ to $A-d\lambda$. A null Killing direction requires a different decomposition.
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