Good behaviour of Lie bracket at a superdensity point of the tangency set of a submanifold with respect to a rank 2 distribution
The paper under review points out an interesting aspect of the computation of the bracket of two vector fields \(H\) and \(K\) of class \(C^1\) on an open subset \(U\) of \(\mathbb{R}^{N+m}\). Namely, for any \(N\)-dimensional \(C^1\)-submanifold \(\mathcal{M}\) of \(U\), a superdensity point of a subset \(E\subseteq\mathcal{M}\) is a point \(z_0\in\mathcal{M}\) satisfying \(\mathcal{H}^N(B_{\mathcal{M}}(z_0,r)\setminus E)=o(r^{N+1})\) as \(r\to0+\), where \(\mathcal{H}^N\) is the \(N\)-dimensional Hausdorff measure. For instance, every interior point of \(E\) in the relative topology of \(\mathcal{M}\) is a superdensity point of \(E\), but there may be many other examples of superdensity points. With this terminology, the main result of this paper is that if \(z_0\in\mathcal{M}\) is a superdensity point of the set \(\mathcal{T}:=\{z\in \mathcal{M}\mid H(z),K(z)\in T_z\mathcal{M}\}\), then \([H,K](z_0)\in T_{z_0}\mathcal{M}\). This fact is quite interesting on its own when \(z_0\) is not an interior point of the set \(\mathcal{T}\) and moreover it allows one to recover a result from one of the author's earlier papers [Indiana Univ. Math. J. 68, No. 2, 393--412 (2019; Zbl 1427.58001)].
tangency set of a submanifold with respect to a distribution
scientific article; zbMATH DE number 7364758
scientific article