Integrability on direct limits of Banach manifolds (Q2183749)

The authors study various kinds of objects in the framework of direct limits of anchored Banach bundles over direct limits of Banach manifolds. More specifically, they obtain a criterion of integrability of a distribution \(\Delta\) on a convenient manifold \(M\) satisfying a number of properties involving an ascending sequence of anchored Banach bundles \((E_n, \pi_n, U_n, \rho_n)_{n\in\mathbb{N}^*}.\) In particular, they assume that for each \(n,\) there exists an almost Lie bracket \([\cdot,\cdot]_n\) on \((E_n, \pi_n, U_n, \rho_n)\) such that \((E_n, \pi_n, U_n, \rho_n, [\cdot,\cdot]_n)\) is a Banach Lie algebroid. Moreover, they show that each maximal integral manifold of the integrable distribution \(\Delta\) satisfies the direct limit chart property at any point and is Hausdorff if \(M\) is Hausdorff. To make the paper self-contained, the authors spend a number of sections to recall various notions: the convenient differential calculus setting of Frölicher, Kreigl and Michor, and the direct limits of various algebraic and geometric objects.