  The recent extensive work on different approaches to the Schottky problem has
produced marked progress on several fronts. At the same time, it has become
apparent that there exist very close connections between the various
characterizations of Jacobian varieties described in Mumford's classic lectures
{\it Curves and Their Jacobians\/}. Until now, the approach via double
translation manifolds has seemed to be quite different from other approaches to
the Schottky problem. The purpose of this paper is to bring this last approach
``into the fold'' as it were, and to show precisely how it relates to
characterizations of Jacobians based on trisecants and flexes of the Kummer
variety, and the K.P. equation.
