Normal forms for Lie algebroids (Q2782562)

A Lie algebroid is a natural generalization of the notion of the Lie algebra and also of the tangent bundle of a manifold. A Lie algebroid is a vector fibre bundle \(A,\) fibered over a base manifold \(M,\) endowed with two structures NEWLINENEWLINENEWLINEa) a Lie bracket structure \([,]\) on the space \(\Gamma A\) of sections of \(A\), NEWLINENEWLINENEWLINEb) a \({\mathcal C}^{\infty }(M)\)-linear mapping \(\#: \Gamma A \mapsto {\mathcal X}(M)\), the Lie algebra of vector fields, which is called the anchor of the Lie algebroid, NEWLINENEWLINENEWLINEwhich satisfy the following properties NEWLINENEWLINENEWLINEc) \(\#\) is a Lie algebra homomorphism, NEWLINENEWLINENEWLINEd) \([\alpha , f\beta]=\#\alpha (f)\beta +f[\alpha , \beta], \forall \alpha , \beta \) sections and \(f \in {\mathcal C}^{\infty }(M).\) NEWLINENEWLINENEWLINEIn the present paper a local splitting theorem for the Lie algebroids is given, showing that \(A\) decomposes locally as the direct product of a tangent algebroid and an algebroid with 0-rank at the origin. NEWLINENEWLINENEWLINEA leaf \(F\) of the Lie algebroid \(A\) is, by definition, a maximal integral submanifold of the distribution \(Im \#.\) If \(N\) is any transversal manifold to \(F\) at a point \(m \in F,\) the transversal algebroid \(A_N\) is defined by the formula \((A_N)_m=\{\alpha \in A_m\mid \#\alpha \in T_mN\}.\) The unicity of the transversal structures is proved. It is shown that \(A_N\) is, up to an algebroid isomorphism, independent of the chosen transversal \(N,\) which passes to \(m.\) The splitting theorem, investigated in the last section, reduces the local study of Lie algebroids to the case where the rank is zero, for which the author gives an analytic linearization for rank 0 algebroids.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].