Dorfman connections and Courant algebroids

Linear connections are useful for describing the tangent spaces of vector bundles, especially their Lie algebroid structure. The direct sum of the tangent space and the cotangent space of a manifold carries the structure of a "standard Courant algebroid'', which naturally extends the Lie algebroid structure of the tangent space. In geometric mechanics, it is often useful to understand the standard Courant algebroid over a vector bundle (e.g. a phase space $T^*Q$). I will introduce the notion of "Dorfman connection'' and explain how the standard Courant algebroid structure over a vector bundle is encoded by a certain class of Dorfman connections. If time permits, I will give more examples, showing that Dorfman connections are natural objects in the study of Courant algebroids and Dirac structures.