Invariants for subspaces of product manifolds

Question

The answer to the following question is probably known since long ago, although unknown to me, since I am not a differential geometer.

Let $X$ and $Y$ be 2-dimensional, smooth manifolds and let $Z$ be an open piece of a hypersurface in $X\times Y$ near a point $(x_0, y_0)$ with the properties that both projections $\pi_X:\ Z \to X$ and $\pi_Y:\ Z \to Y$ have surjective differentials and the projections from the conormal $N^*(Z)$ into $T^*(X)$ and $T^*(Y)$ are local diffeomorphisms. What are the invariants of $Z$ with respect to separate diffeomorphisms in $X$ and $Y$? In particular, how can we decide, for instance in terms of a defining function $F(x_1, x_2, y_1, y_2)$ for $Z$, whether there are coordinate systems in $X$ and $Y$ such that the fibers $\pi_X(\pi_Y^{-1}(y))$ and $\pi_Y(\pi_X^{-1}(x))$ are straight lines (which is the case if $F(x, y) = x_1 y_1 + x_2 + y_2$).

Answer 1

There is a theory due to Tresse, which is unfortunately fairly complicated. It is explained, at least partly, in the book of Arnol'd, Geometric Methods in the Theory of Ordinary Differential Equations. Elie Cartan wrote a difficult paper on it, and this paper was explained more clearly in a paper of Bryant, Griffiths and Hsu, Toward a Geometry of Differential Equations. The possibility of finding coordinates in which both systems of curves are straight lines is precisely the vanishing of Tresse's invariants, which occurs just when your space $Z$ is given by a single quadratic equation in the space $X \times Y$, in some system of coordinates. Explicit formulae for Tresse's invariants are in the sources I mentioned. I wrote a paper on the relation of this problem to complex algebraic geometry: http://arxiv.org/pdf/math/0507087v5.