suppose we have a surface (2-dimensional submanifold of $\mathbb{R}^3$) equipped with a Riemannian metric which is invariant with respect to rotations about the $z$-axis (that is, all rotations about the z-axis are isometries of the surface).
I am wondering if the surface is necessarily invariant with respect to reflections at (Euclidean) planes containing the $z$-axis (e.g. the x-z-plane or y-z-plane).
Surfaces of revolution with the Riemannian metric induced by the space $\mathbb{R}^3$ have this property but I am not sure if it must hold in general.
Best wishes
Yes. For any point $p$ in your surface, its reflection $p'$ at a plane containing the $z$-axis can also be obtained by some rotation around the $z$-axis, hence by the assumption of rotational invariance (under arbitrary rotation angles) $p'$ is also in the surface.