Interpolation of geometric structures compatible with a pseudo Riemannian metric

Let ( M , g ) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g : two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is paracomplex and symmetric with respect to g , then r induces a pseudo Riemannian product structure on M . Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and ω ( x , y ) = g ( j x , y ) is symplectic, then M is almost pseudo Kähler. Now, product, complex and symplectic structures on M are trivial examples of generalized (para)complex structures in the sense of Hitchin. We use the latter in order to define the notion of interpolation of geometric structures compatible with g . We also compute the typical fibers of the twistor bundles of the new structures and give examples for M a Lie group with a left invariant metric.