Local Existence of Chebyshev Nets My Question:
How can one locally guarantee the existence of a Chebyshev net on an arbitrary regular surface?
Additional Information:
In
Existence and construction of Chebyshev nets and application to gridshells, Yannick Masson, page 21
the author claims that the Chebyshev nets locally exist for all regular surfaces and for the proof he cites
Lezioni di geometria differenziale by Bianchi.
I couldn't find the proof in the mentioned book (possibly due to not knowing Italian).
Therefore, I would appreciate if the proof or a reference to a possible proof is given.
 A: The idea is relatively simple. Take the Chebyshev equation $\omega_{xy}=-K\cdot \sin\omega$ and prove that the solutions exist locally given "some" boundary conditions. If you put Cauchy conditions on a curve ($\omega$ and its derivative component normal to the curve) such that the characteristics of the equation (which are the $x$ and $y$ constant lines) intersect the curve once and only once, then the solution exists around the curve because the equation is hyperbolic at every point, as guaranteed by classical theorems like Cauchy–Kowalevski. You can check the details, for instance here sec. 2. If you use Goursat boundary values (instead of Cauchy's) you can use something called Hazzidakis formula to write the solution as a formal convergent series (I think Yannick Masson does that in his PhD thesis, or Étienne Ghys in Sur la coupe des vêtements, whose translation I don't know if exist). There may be some other types of boundary/initial conditions where you can prove existence; let me just mention that the Chebyshev equation is not per-se used, but an equivalent system called Servant equation  (check for instance Equilibrium of Tchebyschev nets by Pipkin 1983).
By the way, why are you interested in Chebyshev nets ?
