Lie algebroid foliations and ¹(M)-Dirac structures

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a ¹(M)-Dirac structure L induces on every leaf F of its characteristic foliation a ¹(F)-Dirac structure L_F, which comes from a precontact structure or from a locally conformal presymplectic structure on F. In addition, we prove that a Dirac structure on M × can be obtained from L and we discuss the relation between the leaves of the characteristic foliations of L and .