This is also the case with the curve $(N,-\mu_{eff})$ with $\varphi_0$ constant. This is a direct consequence of theorem I of  [@sorkin1982].

Here we describe the meaning of criterion (i) in the context of catastrophe theory. We regard the ADM energy $m$ as a function of three variables $\varphi_0$, $N$, and $\omega$. We take $\omega$ as a state variable. We consider an equilibrium space $$M_m=\left\{(\varphi_0,N,\omega)
      \Biggm|\left(\frac{\partial m}{\partial \omega}
    \right)_{\varphi_0,N}=0\right\}$$ and a control space $${\bf R}^2=\{(\varphi_0,N)\}.$$ We define a catastrophe map $$\begin{aligned}
  \chi_m : M_m~ & & \longrightarrow~ {\bf R}^2, \nonumber \\ 
           (\varphi_0, N, \omega)~ & & \longmapsto ~(\varphi_0,N).  \end{aligned}$$ A point $P \in M_m$ is called a singular point of $\chi_m$ if the Jacobian of $\chi_m$ vanishes at $P$. A point $Q \in {\bf R}^2$ is called a singular value if there is at least one singular point in $\chi_m^{-1}(Q)$. A bifurcation set $B_m\subset {\bf R}^2$ is a set of singular values. At a singular point $P\in M_m$, a vector normal the tangent space of $M_m$, which is $$\left(\left(\frac{\partial^2 m}
      {\partial\varphi_0\partial\omega}\right)
  _{N,\omega},
  \left(\frac{\partial^2 m}{\partial N\partial\omega}\right)
  _{\omega,\varphi_0},
  \left(\frac{\partial^2 m}{\partial\omega^2}\right)
  _{\varphi_0,N}\right),$$ is parallel to the $\varphi_0 N$ plane. Therefore, the set of singular points, $\Sigma_m\subset M_m$, satisfies $$\Sigma_m=\left\{(\varphi_0,N,\omega)\Biggm|
      \left( \frac{\partial m}{\partial \omega}
        