Integrability of weak distributions on Banach manifolds

Abstract

This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroids and Banach Poisson manifold. The main results of this paper generalize the works presented in [ChSt], [Nu] and [Gl].In differential geometry, a distribution on a smooth manifold M is an assignment D :. On the other hand, D is called involutive if, for any vector fields X and Y on M which are tangent to D, the Lie bracket [X, Y ] is also tangent to D. The distribution is invariant if for any vector field X tangent to D, the flow φ X t leaves D invariant (see 2.1).On finite dimensional manifod, when D is a subbundle of T M , the classical Frobenius Theorem gives an equivalence between integrability and involutivity. In the other case, the distribution is "singular" and even under assumptions of smoothness on D, in general, the involutivity is not a sufficient condition for integrability (we need some more additional local conditions). These problems were clarified and resolved essentially in [Su], [St] and [Ba].In the context of Banach manifolds, the Frobenius Theorem is again true, for distributions which are complemented subbundles in the tangent bundle. For singular distributions, some papers ([ChSt], [Nu] for instance) show that, when the distribution is closed and complemented (i.e. D x is a complemented Banach subspace of T x M ), we have equivalence between integrability and invariance. Under sufficient conditions about local involutivity we also get a result of integrability. A more recent work ([Gl]) proves analog results without the assumption that the distribution is complemented.According to the notion of "weak submanifolds" in a Banach manifold introduced in (([El],[Pe]), in this paper, we consider "weak distributions": D x can be not closed in T x M but D x is endowed with its own Banach structure, so that the inclusion D x → T x M is continuous. Such a category of distributions takes naturally place in the framework of Banach Lie algebroids (morphisms from a Banach bundle over a Banach manifold into the tangent bundle of this manifold). Under conditions

Snakes and articulated arms in an Hilbert space

“…The arguments used in our proof can be found in [5] and [4]. Moreover, this Theorem of accessibility can be seen as an application of results obtained in [4]; it also gives rise to an illustration of the "almost Banach Lie algebroid structures" developed in [1] (see Appendix 5).…”

“…• According to [5], a weak distribution on M is an assignment D : x → D x which, to every x ∈ M , associates a vector subspace D x in T x M (not necessarily closed) endowed with a norm || || x such that (D x , || || x ) is a Banach space (denoted byD x ) and such that the natural inclusion i x :D x → T x M is continuous. Moreover, if the Banach structure on D x is a Hilbert structure, we say that D is a weak Hilbert distribution.…”

“…Let N be the union of all integral manifolds i L : L → M through x 0 . Then i N : N → M is the maximal integral manifold of through x 0 (see Lemma 2.14 [5]). Proposition 2.3 (see [4]).…”

“…This paper is organized as follows. Section 2 contains all definitions and results of [5] and [4] which are used in the proof about the accessibility sets. At first, the reader can go directly to the section 3 , and he may refer to this paragraph 2 only for a more detailed reading.…”

Dirac Structures on Banach Lie Algebroids

“…They find conditions for a Dirac structure to be a Banach Lie algebroid. The notion of Banach Lie algebroid was introduced by M. Anastasiei, [1] and independently by F. Pelletier [10].…”