Lie algebroid foliations and ${\cal E}^1(M)$-Dirac structures

Details Lie algebroid foliations and ${\cal E}^1(M)$-Dirac structures Lie algebroid foliations and ${\cal E}^1(M)$-Dirac structures

2001-06-11

arxiv.org/abs/math/0106086v1

Mathematics

Differential Geometry

25 pages

Scientific paper

10.1088/0305-4470/35/18/307

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 ${\cal E}^1(M)$-Dirac structure $L$ induces on every leaf $F$ of its characteristic foliation a ${\cal E}^1(F)$-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 $\tilde{L}$ on $M\times \R$ can be obtained from $L$ and we discuss the relation between the leaves of the characteristic foliations of $L$ and $\tilde{L}$.

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