A model form for exact $b$-metrics

Any manifold with boundary can be equipped with a $b$-metric which takes the form $\frac{dx^2}{x^2} + h(x,y,dx,dy)$ with respect to some product decomposition near the boundary, and $h$ positive definite on restriction to the tangent space of the boundary. Here we show the existence of a product decomposition such that $h$ is independent of $dx$ modulo terms vanishing to infinite order at the boundary. The uniqueness of this decomposition is also examined.