# DOFs in coordinate mapping if metrics specified?

## Main Question or Discussion Point

I am trying to determine a mapping between two coordinate systems, given only the metric tensor written for each system. How much freedom do I have available in specifying the mapping once the metrics have been given?

I know that the transformation equations of the metric tensor components provides n*(n+1)/2 conditions, but I am trying to figure out if there are other restrictions that mapping must meet. In other words, do I have n^2-n*(n+1)/2 free functions that can be specified to determine the partial derivatives of the transformation (e.g., \frac{\partial x'}{\partial x}, etc.)?

I have been wondering whether the curvature tensor places a set of restrictions. For instance, do the second order partial differential equations for the change in coordinates (which involve the Christoffel symbols of the second kind, and hence the metric tensor derivatives) provide a further restriction consistent with ensuring that the curvature is the same when determined by the metric tensor for each of coordinates? If so, these equations are field equations that require boundary conditions. Do I have the freedom to describe the boundary mapping arbitrarily?