Does somebody have some interesting examples of Courant algebroids which are not exact? By exact I mean one which is of the form $TM\oplus T^\star M$ with the standard bracket twisted by a closed 3-form $H$.


Thank you!



Courant algebroids which are not exact
The answer depends on what you mean by ``interesting."


For example the paper "On the Geometric Structure of Hamiltonian Systems
with Ports" by 
Jochen Merker, J Nonlinear Sci (2009) 19: 717–738
DOI [10.1007/s00332-009-9052-3](http://link.springer.com/article/10.1007%2Fs00332-009-9052-3), may be considered as dealing with interesting examples of Courant algebroids. The algebroids there are not of the form $TM\oplus T^\*M \to M$.


Grützmann's thesis ([arXiv:1004.1487 [math.DG]](http://arxiv.org/abs/1004.1487)) maybe another good place to look.


