Finding the curvature of a distorted manifold
I have a distorted 2D manifold, with Cartesian coordinate system (u,v).
This has an automorphism with an undistorted manifold with Cartesian coordinates (x,y).
The automorphism is defined by a vector field (delta_x(x,y), delta_y(x,y)), where delta_x gives the displacement values (u - x) over the manifold, and similarly delta_y the displacement values (v - y).
My question is how do I find a scalar field F1(u,v), representing the curvature of the distorted manifold with respect to the x displacements given by delta_x(x,y).
As I recall the curvature of a scalar field is normally given by the Ricci Scalar; but that's too difficult to calculate, so I need a simpler way of determining the distortion at any given point. (It can be an approximate measure; it doesn't have to be precise.)