Double Products

are closed. Manifolds carrying a hypersymplectic structure have a rich geometry, the neutral metric is Kähler and Ricci flat and its holonomy group is contained in Sp(2n,R) ([8]). Moreover, the Levi Civita connection is flat, when restricted to the leaves of the canonical foliations associated to the product structure given by E (see [2]). Metrics associated to a hypersymplectic structure are also called neutral hyperkähler (see [10]). Hypersymplectic structures have significance in string theory. In [14], N = 2 superstring theory is considered, showing that the critical dimension of such a string is 4 and that the bosonic part of the N = 2 theory corresponds to self-dual metrics of signature (2, 2) (see also [5] and [9]). The quotient construction proved to be a powerful method to construct symplectic and hyperkähler structures on manifolds. According to [8] this method cannot be always applied in the setting of hypersymplectic structures. Compact complex surfaces with neutral hyperkahler metrics are biholomorphic to either complex tori or primary Kodaira surfaces and both carry non flat neutral hyperkahler metrics, by results of Kamada (see [10]). In higher dimensions, hypersymplectic structures on a class of compact quotients of 2-step nilpotent Lie groups were exhibited in [6] in their search of neutral Calabi-Yau metrics. The purpose of this paper is to give a procedure to construct hypersymplectic structures on R4n with complete and not necessarily flat associated neutral metrics. The idea behind the construction will be to consider the canonical flat hypersymplectic structure on R4n and then translate it by using an appropriate group acting simply and transitively on R4n. This group will be a double Lie group (R4n, R2n × {0}, {0} × R2n) constructed from affine data on R2n. The most important feature achieved by this procedure is that the associated neutral metrics