On equivalence of simple closed curves in flat surfaces

By explicit constructions, we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g > 1, there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that l _X (α) ≠ l _X (β), ext _X (α) ≠ ext _X (β) and l _q (α) ≠ l _q (β), where l _X (·), ext _X (·) and l _q (·) are the hyperbolic length, the extremal length and the quadratic differential length respectively. These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.