I'm a physics student and most of the mathematics that I think I know I learned self-taught. That said, I have the following question that may be very naive but I don't know where to really ask.
If I have a manifold gifted with the metric, $$ ds^2 = d^2 r + d^2 \varphi$$ which comes from a larger manifold that has spherical symmetry. I know that locally a manifold is isomorphic to a ball in $\mathbb{R}^n$, so, I can configure a Cartesian coordinate system with axes $x_1$ and $x_2$ such that to describe points far from the established origin You can apply a limit by parameterizing $x_1$, $x_2$ using: $x_1 = r \cos \varphi$, $x_2 = r \sin \varphi$ and recover spherical symmetry? or is it something that cannot be done because I am starting from a local property?