Integrability of weak distributions on Banach manifolds
type: Item
instance of: scholarly article
title: Integrability of weak distributions on Banach manifolds
zbMATH Open document ID: 1286.46087
author: F. Pelletier
published in: Indagationes Mathematicae. New Series
publication date: 2012-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.1950
Mathematics Subject Classification ID: 46T05
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53D17
zbMATH DE Number: 6070478
zbMATH Keywords: Banach manifold
zbMATH Keywords: weak Banach submanifold
zbMATH Keywords: weak distribution
zbMATH Keywords: integral manifold
zbMATH Keywords: involutive distribution
zbMATH Keywords: integrable distribution
zbMATH Keywords: invariance
zbMATH Keywords: Lie invariance
zbMATH Keywords: Banach Lie algebroid
zbMATH Keywords: Banach Poisson manifold
MaRDI profile type: Publication
OpenAlex ID: W2047214671
arXiv ID: 1012.1950
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DOI: 10.1016/J.INDAG.2011.11.005
review text: The article is devoted to infinite-dimensional generalizations of the Stefan-Sussmann theorem in the context of Banach manifolds, and related topics; see [\textit{P. Stefan}, J. Lond. Math. Soc., II. Ser. 21, 544-556 (1980; Zbl 0432.58002)] and [\textit{H. J. Sussmann}, Trans. Am. Math. Soc. 80, 171--188 (1973; Zbl 0274.58002)] for the finite-dimensional case, with corrections in [\textit{R. Balan}, Boll. Unione Mat. Ital., VII. Ser., A 8, No. 3, 335--344 (1994; Zbl 0852.57024)].NEWLINENEWLINE Let \(M\) be a connected smooth manifold modelled on a Banach space. A weak distribution on \(M\) is a family \(D:=(D_x)_{x\in M}\) of vector subspaces \(D_x\subseteq T_xM\) of the tangent space at \(x\in M\), together with norms \(\|.\|_x\) on \(D_x\) such that \(\tilde{D}_x:=(D_x,\|.\|_x)\) is a Banach space and the inclusion map \(i_x\colon \tilde{D}_x\to T_xM\) is continuous. If \(D_x\) is a closed vector subspace of \(T_xM\) for each \(x\in M\), then the distribution is called closed.NEWLINENEWLINE A weak submanifold of \(M\) is a pair \((N,f)\) of a Banach manifold \(N\) and an injective map \(f\colon N\to M\) which is a quasi-immersion, i.e., the tangent map \(T_xf\colon T_xN\to T_{f(x)}M\) is injective for all \(x\in N\). After replacing \(f\) with its image, one may always assume that \(N\subseteq M\) and \(f\) is the inclusion map. A weak submanifold \(N\subseteq M\) is called an integral manifold for a weak distribution \(D\) on \(M\) through \(x\in M\) if \(x\in N\) and \(T_yN=D_y\) for all \(y\in N\) (with the topology from \(\|.\|_y\)).NEWLINENEWLINE A weak distribution \(D\) on \(M\) is called integrable if for each \(x\in M\) there exists an integral manifold \(N\subseteq M\) through \(x\). Finally, a weak distribution \(D\) on \(M\) is said to be lower trivial if, for each \(x\in M\), there exists an open neighborhood \(V\subseteq M\) of \(x\) and a smooth map \(\Theta\colon \tilde{D}_x \times V\to TM\) (called a lower trivialization) such that \(\Theta_y:=\Th…
reviewed by: Helge Glöckner
description: scientific article; zbMATH DE number 6070478