I'm trying to understand the manifold properties of world-sheets in string theory. I'm told that world sheets are manifolds and that manifolds are locally Euclidean. So I would like to know the characteristics between the space-time coordinates of the world-sheet given as xμ verses the 2D surface parameterized by (σ,τ). Are xμ locally Euclidean? Are the coordinates (σ,τ) locally Euclidean? Remember xμ are functions of the parameters (σ,τ) or xμ=xμ(σ,τ) which defines a surface in space-time. How does this all relate to manifold theory?