Courant-Nijenhuis tensors and generalized geometries

Details Courant-Nijenhuis tensors and generalized geometries Courant-Nijenhuis tensors and generalized geometries

2006-01-31

arxiv.org/abs/math/0601761v1

Monograf\'ias de la Real Academia de Ciencias de Zaragoza 29 (2006), 101--112

Mathematics

Differential Geometry

10 pages

Scientific paper

Nijenhuis 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.

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