  Using Monge-Amp\`ere geometry, we study the singular structure of a class of
nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical
fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by
examining the role of an induced metric on Lagrangian submanifolds of the
cotangent bundle. In particular, we show that the signature of the metric
serves as a classification of the Monge-Amp\`ere equation, while singularities
and elliptic-hyperbolic transitions are revealed by the degeneracies of the
metric. The theory is illustrated by application to an example solution of the
semigeostrophic equations.
