2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/121441Nijenhuis tensors $N$ on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form $N+N^*=aI$ for irreducible Courant algebroids, in particular for the extended tangent bundles $TM\oplus T^*M$. It is proved that compatible Nijenhuis tensors on irreducible Courant algebroids must satisfy quadratic relations $N^2-aN+bI=0$, so that the corresponding hierarchy is very poor. The particular case $N^2=-I$ is associated with Hitchin's generalized geometries and the cases $N^2=I$ and $N^2=0$ -- to other "generalized geometries". These concepts find a natural description in terms of supersymplectic Poisson brackets on graded supermanifolds.10 pagesDifferential GeometryMathematical Physics17B99, 17B62, 53C15, 53C56, 53D05, 53D17text