Let $f(s,t)$ be a two times continuously differentiable piece of a surface, which is part of a Riemannian manifold $M$.
Let $0\leq s\leq 1$ and $-\epsilon<t<\epsilon$ be given such that all $s$-curves are geodetic lines and unit speed curves for fixes $t_0$.
Suppose, every curve of these subtends the $t$-line $f(0,t)$ in $t=t_0$ orthogonally.
Question: Why does it follow that the $s$-lines and $t$-lines subtend one another orthogonally in every intersection point?
Thanks for the help.