Integrability of weak distributions on Banach manifolds
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 \(\Theta_y:=\Theta(.,y)\) is a linear map from \(\tilde{D}_x\) to \(T_yM\) for each \(y\in V\) and there exists a linear topological embedding \(\tilde{\Theta}_y\colon \tilde{D}_x\to \tilde{D}_y\) such that \(i_y\circ \tilde{\Theta}_y=\Theta_y\) and \(\tilde{\Theta}_x\) is the identity on \(\tilde{D}_x\). Let \(X_D\) be the set of all smooth vector fields \(X\) on open subsets of \(M\) such that \(X(x)\in D_x\) for each \(x\) in the domain of \(X\). Let \(X_D^-\) be the set of all vector fields of the form \(X=\Theta(u,.)\) for some lower trivialization \(\Theta\) as before and \(u\in \widetilde{D}_x\).NEWLINENEWLINE The author shows: A lower trivial weak distribution \(D\) on a Banach manifold \(M\) is integrable if and only if \(D\) is \(X_D^-\)-invariant (in the sense that the tangent maps \(T\Phi_t\) of the flow of \(X\in X_D^-\) take the spaces \(D_x\) onto each other). In this case, there is a maximal connected integral manifold through each point \(x\in M\) (Theorem 1). The author also recovers (as Corollary 2.10) a result which previously arose in a joint project of the reviewer and Rezső Lovas (and will be made available in a joint paper with Jan Milan Eyni): A lower trivial closed distribution \(D\) on a Banach manifold \(M\) is integrable if and only if it is \(X_D\)-invariant (if and only if it is \(X_D^-\)-invariant); for the case of complemented closed vector subspaces \(D_x\subseteq T_xM\), see already [\textit{D. Chillingworth} and \textit{P. Stefan}, Math. Proc. Camb. Philos. Soc. 79, 117--128 (1976; Zbl 0317.58002)].
scientific article; zbMATH DE number 6070478
scientific article