I'm trying to find any reference that defines what a "conjugate direction field" is. I think this is a topic of differential geometry, but I can't find any reference for this, I have many papers that mention this concept but none of them provides a good definition or reference I can look up.
Is there any reference you can suggest to understand the topic?
The context with which I am familiar is this: Given a surface $S$ in $\Bbb R^3$, its second fundamental form $\mathbf{II}$ defines a symmetric bilinear form on each tangent space. We say tangent vectors $v$ and $w$ at $p\in S$ are conjugate if $\mathbf{II}_p(v,w)=0$. (So, on a round sphere, this would just be saying that they're orthogonal in $\Bbb R^3$.)
For an interesting application of this, see Exercise 20 on p. 77 of my differential geometry text.