From the Levi-Civita connection to Levi-Civita connections: Effective actions

To every metric manifold, there is an associated unique linear connection playing a fundamental role in the physics of gravity.
Einstein-Hilbert action then allows one to encode the Einstein equations in terms of a simple action functional. Various
generalizations of this idea appear throughout the string and supergravity theory. Is there a similar geometrical content in
more general cases? 

Some answers can be given in terms of Levi-Civita connections on Courant algebroids. The definition and basic examples of Courant
algebroids will be recalled, and the suitable generalization of linear connections, torsion and curvature tensors will be presented.
We will show how a bosonic field content of a Type II supergravity can be encoded in terms of a generalized Ricci tensor and the
corresponding scalar curvature. Possible applications of this viewpoint will be sketched.