Sorry for having such a long and confusing title.
The question arose while reading Thermodynamic ensembles and gravitation by J D Brown et al (https://dx.doi.org/10.1088/0264-9381/7/8/020).
In Section 3, the authors introduce a metric of the form $$ \newcommand{\dif}{\mathrm d} \dif s^2 = \beta^2 \dif \tau^2 + \alpha^2 \dif y^2 + \frac{A}{2\pi} (\dif \theta^2 + \sin^2 \theta \,\dif \varphi^2) $$ where $\alpha, \beta, A = 4\pi r^2$ are functions of $y \in [0, 1]$, and $\tau$ is $2\pi$-periodic. In my understanding, $y$ acts as a form of “radial coordinates”.
The authors say:
The centre $y = 0$ is not an element of the boundary (which is three dimensional), but rather is a regular $2$-surface in $M$, with non-zero area $A(0) = 4\pi r_+^2$. Thus the centre is distinguished as a degenerate leaf of the radial foliation. The $y$-$\tau$ plane near $y = 0$ must be isometric to a flat disc. The regularity condition for the metrics at $y = 0$ is $$ \alpha^{-1} \beta' |_{y = 0} = 1 $$ The prime meaning $\frac{\partial}{\partial y}$.
By “boundary”, the authors mean $y = 1$.
The whole paragraph does not make sense to me. Normally when we say the regularity condition for $f(y)$ at centre is something like $f(r) < \infty$, or $f'(0) = 0$.
Can anyone give me some hints about how does the “isometric to flat disc” stuff work?
