What I am looking for a method to attach a 'coordinate system' to a 2D surface (embedded in $\mathbb R^3$) honoring certain constraints to the Jacobian.
To give context it is about generating texture coordinates in computer graphics.
For the coordinates all of $\mathbb R^2$, $\mathbb R^3$, $\mathbb R^4$ would be useful. They need only be defined on the surface. More precisely these are a mapping from the manifold of the surface to $\mathbb R^n$ vectors.
The constraints one would give are likely over-defined, with no exact solution. Exact solutions are not needed, though, so the task should probably by formulated as an optimization problem.
Optimization targets would be of two forms:
global constraints to the Jacobian (maximal smoothness, orthogonality, iso-lines following manifold geodesics)
local constraints (defined on manifold points or patches) requiring vector components of the Jacobian to have certain direction and/or vector length (defining direction and stretch of textures)
I believe the solution should ideally not on the embedding of the surface in $\mathbb R^3$, and thus be formulated by treating the surface as a Riemannian manifold.
I try to find out which methods exist to solve problems similar to the one sketched above. They should be stable (insensitive to small changes to the manifold geometry and local constraints) and efficiently numerically solvable.
Unfortunately I don't know what exactly I should search for (subfield of mathematics, keywords). Can someone point me in the right direction? The best description I could come up with is 'numerical methods for partial differential equations on Riemannian manifolds' which is of course way to broad (any maybe not even correct).