Commun. Math. Phys. 188, 305 – 325 (1997) Communications in Mathematical Physics c  Springer-Verlag 1997 On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy Franco Magri 1 , Marco Pedroni 2 , Jorge P. Zubelli 3 1 Dip. di Matematica, Universit` a di Milano, Via C. Saldini 50, Milano MI 20154, Italy 2 Dip. di Matematica, Universit` a di Genova, Via Dodecaneso 35, Genova GE 16146, Italy 3 IMPA – CNPq, Est. D. Castorina 110, Rio de Janeiro RJ 22460, Brazil Received: 23 July 1996 / Accepted: 6 January 1997 Abstract: We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. 1. Introduction The theory of Darboux transformations has a long and curious history. These transfor- mations were introduced more than a century ago by G. Darboux in [8] and after passing through a period of oblivion, they were rediscovered as a technique for constructing solutions to important partial differential equations of Mathematical Physics. They have been used in a number of interesting situations, as can be seen in [1, 9, 10, 17, 18, 23, 24] and references therein. The “philosophical” purpose of the present work is to show that the Darboux tech- nique has a deeper geometric significance than the usually accepted explanation that the underlying equations are “covariant” by a certain set of formal manipulations [17]. Furthermore, we argue below that an appropriate geometric setting of the Darboux method allows to clarify the links among important elements of Soliton Theory, to wit the Kadomtsev-Petviashvili (KP) hierarchy, the modified KP hierarchy, the (generalized) Miura map, and the (generalized) Toda lattice. We do so, by approaching the Darboux method as a theory of intertwining of vector fields, as follows: