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 3form $H$.
Thank you!
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 3form $H$. Thank you! 


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/s0033200990523, 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]) maybe another good place to look. 

