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.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3946