  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.
