Suppose you have a 2-dimensional surface embedded into a 3-dimensional ambient space by the map $R:\mathbb{R}^2\to\mathbb{R}^3$. Suppose that for every point on the surface, you could assign a 3-dimensional basis vector frame.
Is it safe to say that $\textbf{since}$ the 3-dimensional basis is strictly restricted to every point on the surface, and that the surface contains $\textbf{only}$ a subset of the points of $\mathbb{R}^3$...then the 3-dimensional basis defined on the surface must be $\underline{\text{non-holonomic}}$ since it cannot be extended globally over all of $\mathbb{R}^3$?
And as a follow up question...does this mean that for tensor and vector fields defined using this basis on the surface, there will be some sort of torsion present in the basis' connection that appears only for non-holonomic bases?