Journal of the
Korean Mathematical Society JKMS

We consider  a vector bundle map $F\colon E_{1}\longrightarrow E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$  over arbitrary bases $M_{1}$ and  $M_{2}$. We associate to it different notions of curvature which we call  A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten  structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and  we will prove that these curvatures are equivalent, in the  sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary  that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten   supermanifolds).