Semi-simple generalized Nijenhuis operators (Q1951051)

From the point of view of the Courant (or generalized) geometry, the usual tangent bundle of a manifold \(M\) must be replaced by the generalized tangent bundle \(\mathcal{T}M:=TM\oplus T^*M\). In this paper the authors consider a special class of endomorphism fields of \(\mathcal{T}M\) that have a vanishing Courant-Nijenhuis torsion and is diagonalizable (after a possible extension of scalars) with constant dimensions of its eigenspaces. Such an endomorphism field is called a semi-simple generalized Nijenhuis operator. The generalized paracomplex and complex structures give examples of such operators. In this paper, it is proved that a semi-simple generalized Nijenhuis operator with two real eigenvalues and two totally isotropic eigenspaces furnishes a pair of transverse Dirac structures with compatibility conditions. Also, it is proved that any semi-simple generalized Nijenhuis operator with two non-real eigenvalues is affinely related to a generalized complex structure. Another important result is that a semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.