This question was previously posted in Orthogonal differentiable family of curves . But I'm facing unsolved issues.
I'm still interested in solve the following exercise:
QUESTION: We say that a set of regular curves on a surface $S$ is a differentiable family of curves on $S$ if the tangent lines to the curves of the set make up a differentiable field of directions. Assume that a surface $S$ admits two differentiable orthogonal families of geodesics. Prove that the Gaussian curvature of $S$ is zero.
where a differentiable field of directions is:
Definition: A field of direction $r$ in a open $U \cap S$ is a correspondence which assigns to each $p$ $\in$ $U\cap S$ a line $r(p)$ in $T_pS$ passing through $p$. $r$ is said to be diferentiable at $p$ $\in$ $U \cap S$ if there exists a nonzero differentiable vector field $w$, defined in a neighborhood $V\cap S$ $\subset$ $U \cap S$ of $p$, such that for each $q$ $\in$ $V\cap S$, $w(q) \neq 0$ is a basis of $r(q)$; $r$ is diferentiable in $U\cap S$ if it is differentiable for every $p$ $\in$ $U$.
At the end of the book, Manfredo give us the following hint
Manfredo's Hint: Parametrize a neighborhood of $p\in S$ in such a way that the two families of geodesics are coordinate curves (Corollary 1, Sec. 3-4). Show that this implies that $F=0$, $E_v=0, $ $G_u = 0$. Make a change of parameters to obtain that $\bar{F} = 0$, $\bar{E} = \bar{G} =1$.
where Corollary 1, Sec.3-4 is:
Corollary 1. (Sec.3-4): Given two fields of directions $r$ and $r'$ in an open set $U \subset S$ such that at $p$ $\in$ $U$, $r(p) \neq r'(p)$, there exists a parametrization $x$ in a neighborhood of $p$ such that the coordinate curves of $x$ are the integral curves of $r$ and $r'$.
And now my problem appears.
Let $\tau_1$ be the field of directions associated to the first family, and $\tau_2$ be the field of directions associated to the second family (note that $\tau_1$ and $\tau_2$ are orthogonal).
I really don't understand why when we apply the Corollary 1 (Sec. 3-4), on the fields of directions $\tau_1$ and $\tau_2$, we end up getting a chart $x: U \subset \mathbb{R}^2 \rightarrow W \cap S$, in such a way that the coordinate curves, $x(u, c^{te})$ and $x(c^{te},v)$, are the two families of geodesics.
From what I know, when the coordinate curves are the integral curves of fields of directions $\tau_1$ and $\tau_2$, we can only conclude that $x_u(u, c^{te})$ is l.d with the direction $\tau_1(x(u, c^{te}))$ and $x_v(c^{te},v)$ is l.d. with the direction $\tau_2(x(c^{te}, v))$.
GI can't see any reason for these coordinate curves to solve the geodesic's differential equation. In the best case scenario I was only able to conclude that the coordinate curves have the same path of a geodesic (which implies nothing).
Does anyone know how I outline this problem and solve the exercise?