Dorfman connections and Courant algebroids

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection \(\nabla\colon \mathfrak{X}(M)\times\Gamma(E)\to\Gamma(E)\) on a vector bundle \(E\) over a smooth manifold \(M\) is tantamount to a linear splitting \(TE\simeq T^{q_E}E\oplus H_\nabla\), where \(T^{q_E}E\) is the set of vectors tangent to the fibres of \(E\). Furthermore, the curvature of the connection measures the failure of the horizontal space \(H_\nabla\) to be integrable. We show that linear horizontal complements to \(T^{q_E}E\oplus (T^{q_E}E)^\circ\) in the Pontryagin bundle over the vector bundle \(E\) can be described in the same manner via a certain class of Dorfman connections \(\Delta\colon \Gamma(TM\oplus E^*)\times\Gamma(E\oplus T^*M)\to\Gamma(E\oplus T^*M)\). Similarly to the tangent bundle case, we find that, after the choice of a linear splitting, the standard Courant algebroid structure of \(TE\oplus T^*E\to E\) can be completely described by properties of the Dorfman connection. As an application, we study splittings of \(TA\oplus T^*A\) over a Lie algebroid \(A\) and, following Gracia-Saz and Mehta, we compute the representations up to homotopy defined by any linear splitting of \(TA\oplus T^*A\) and the linear Lie algebroid \(TA\oplus T^*A\to TM\oplus A^*\). Further, we characterise VB- and LA-Dirac structures in \(TA\oplus T^*A\) via Dorfman connections.