Assume we have a smooth Riemannian metric $g$ on a small one-sided neighborhood $U$ of $0$ on the plane, say $U_\epsilon=\lbrace(x, y): x^2+y^2<\epsilon, y\geq 0\rbrace$.

When does there exist a diffeomorphism $f$ of ( may be smaller) one-sided neighborhood of zero $U_\delta$ such that the resulting Riemannian metric $f_\star g$ can be smoothly extended $\textbf{by relflection}$ along the axis $y=0$.

In particular( but not only) I am mostly interested in cases:

$a)$ $g=\rho(x,y) g_0$, where $\rho(x,y)$ is a conformal factor and $g_0$ is standard Euclidean metric.

$b)$ Case $a$ with flat $g$, i.e. $\Delta\ln\rho=0$