  We prove local-in-time existence of smooth solutions to the semigeostrophic
equations in the general setting of smooth, bounded and simply connected
domains of $\mathbb{R}^2$ endowed with an arbitrary conformally flat metric and
non-vanishing Coriolis term. We present a construction taking place in Eulerian
coordinates, avoiding the classical reformulation in dual variables, used in
the flat case with constant Coriolis force, but lacking in this general
framework.
