Semi-simple generalized Nijenhuis operators

Abstract

We study a special class of endomorphism fields of the generalized tangent bundle ${\mathcal{T}}M:=TM\oplus T^*M$ of a smooth manifold $M$. An operator of this class is defined as follows: it has 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. &nbsp In this study, we distinguish two cases according to whether the operator has exactly two eigenvalues or has at least three elements in its spectrum. In the first case, we prove that either the operator is affinely related to a generalized complex structure, or it is equivalent to a pair of transverse Dirac structures on ${\mathcal{T}}M$. In the second case, the semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.