Dirac structures and paracomplex manifolds (Q1876869)

This paper defines a generalized almost paracomplex structure on a manifold \(M\) to be a vector bundle automorphism \(I\) of \(TM\oplus T^* M\) with \(I^2=\text{identity}\), \(I\neq\pm\text{identity}\), which is orthogonal with respect to a standard bilinear pairing on the bundle. The eigenbundles of \(I\) become transversal subbundles which are maximal isotropic with respect to the bilinear pairing, and these are called almost Dirac structures. Several standard geometric structures are shown to provide examples.