tensor ${\textbf{k}}$ we take to be the pullback to $\mathcal{U}$ of a tensor on ${\partial}M$ which by abuse of notation we write as ${\textbf{k}}\in Sym^{0,2}({\partial}M)$, having the property that ${\textbf{k}}$ restricted to any fiber is positive definite. The metric $g_0 = dx^2 + h +
x^{2k} {\textbf{k}}$ is an exact incomplete cusp edge metric. Note that in coordinates $(x, y, z)$ such a metric takes the form $$\label{eq:exact-cuspedgemetric}
  g_0 =  
{\left(}\begin{array}{ccc}
  dx & dy^i & x^{k} dz^\alpha
\end{array}
{\right)}{\left(}\begin{array}{ccc}
1 & 0 & 0 \\
0 & (h_{ij}) & x^k ({\textbf{k}}_{i\alpha}) \\
0 & x^k ({\textbf{k}}_{\alpha i}) & {\textbf{k}}_{\alpha \beta}
\end{array}
{\right)}{\left(}\begin{array}{c}
  dx \\ dy^i \\ x^{k} dz^\alpha
\end{array}
{\right)}.$$ Thus an incomplete cusp edge metric diffes from an exact one by a polyhomogeneous error of order $O(x^k)$ in norm and can therefore be taken to correspond uniquely to an exact cusp edge metric.

Note that an exact incomplete cusp edge metric (and thus an incomplete cusp metric) gives rise, locally over the boundary to well defined notions of the operators $d_Y$, $d_Z$, the Hodge star operators $\star_Y$ and $\star_Z$, obtained by allowing $\star_{(Y,h)}$ the Hodge star associated with $(Y,
h)$, the Riemannian structure on the base $Y$, to act 