International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 11, Pages 1737-1758

Transitive Courant algebroids

Izu Vaisman

Department of Mathematics, University of Haifa, Haifa 31905, Israel

Received 28 September 2004; Revised 13 March 2005

Copyright © 2005 Izu Vaisman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum EC where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection. In particular, this class contains all the transitive Courant algebroids of minimal rank; for these, the flat term mentioned above is zero. The results extend to regular Courant algebroids, that is, Courant algebroids with a constant rank anchor. The paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.