  The astrophysical importance of the Kerr spacetime cannot be overstated. Of
the currently known exact solutions to the Einstein field equations, the Kerr
spacetime stands out in terms of its direct applicability to describing
astronomical black hole candidates. In counterpoint, purely mathematically,
there is an old classical result of differential geometry, due to Darboux, that
all 3-manifolds can have their metrics recast into diagonal form. In the case
of the Kerr spacetime the Boyer-Lindquist coordinates provide an explicit
example of a diagonal spatial 3-metric. Unfortunately, as we demonstrate
herein, Darboux diagonalization of the spatial 3-slices of the Kerr spacetime
is incompatible with simultaneously putting the Kerr metric into unit-lapse
form while retaining manifest axial symmetry. This no-go theorem is somewhat
reminiscent of the no-go theorem to the effect that the spatial 3-slices of the
Kerr spacetime cannot be chosen to be conformally flat.
