Kaluza-Klein decomposition along a Killing field ($h=V(dx+A{)}^2+\gamma$)

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 $x^{\prime }=x+\lambda$ sends $A$ to $A-d\lambda$. A null Killing direction requires a different decomposition.

For $h=\alpha d{x}^2+\beta d{y}^2+\chi (dz-ydx{)}^2$, the Kaluza-Klein decomposition along a Killing field ${\partial }_x$ has $V=\alpha +\chi y^2$, $A=-\chi ydz/V$ and $\gamma =\beta d{y}^2+\alpha \chi d{z}^2/V$. For a Riemannian metric, positive constants make the fibre norm and quotient metric positive everywhere.