Invariance of $ds^2$ in Randall-Sundrum models

My question is about the invariance of the space-time interval $ds^2$ under orbifold symmetries, such as in the Randall-Sundrum model.

In this model, the space-time is 5-dimensional with metric

$$ds^2 = e^{-2A(y)}eta_{munu}dx^mu dx^nu – dy^2$$ where $eta_{munu}$ is the usual Minkowski 4D metric and $y$ is the 5-th spatial coordinate.

In this model, the space-time is a circle $S_1$ with antipodal points identified under the $Z_2$ parity $ysim-y$, so that the points linked by the equivalence $y sim y + 2pi$ and $ysim-y$ are identified.

The $Z_2$ symmetry is usually used to say that $A(y)sim |y|$ since $ds^2$ should be invariant under the orbifold symmetry $ysim-y$. By solving the Einstein equations, one get the background solution

$$ds^2 = e^{-2k|y|}eta_{munu}dx^mu dx^nu – dy^2,,qquadqquad (text{Eq}.1)$$

where $k$ is a constant related to the curvature of the AdS space.

What about invariance under the symmetry of the circle $y sim y+2pi $? How is this latter identification encoded in the metric Eq.$(1)$? Is it trivially encoded because of coordinate transformations invariance?