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Keywords:
tangency set; distributions; superdensity; integral manifold; Frobenius theorem
tangency set; distributions; superdensity; integral manifold; Frobenius theorem
Summary:
Let $\Phi _1,\ldots ,\Phi _{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U\subset ^{N+m}$, let $\mathbb{M}$ be a $N$-dimensional $C^k$ submanifold of $U$ and define \[ \mathbb{T}:=\lbrace z\in \mathbb{M}: \Phi _1(z), \ldots , \Phi _{k+1}(z) \in T_z \mathbb{M}\rbrace \] where $T_z \mathbb{M}$ is the tangent space to $\mathbb{M}$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $\mathbb{M}$) of $\mathbb{T}$: If $z_0\in \mathbb{M}$ is a $(N+k)$-density point (relative to $\mathbb{M}$) of $\mathbb{T}$ then all the iterated Lie brackets of order less or equal to $k$ \[ \Phi _{i_1}(z_0),\, [\Phi _{i_1}, \Phi _{i_2}](z_0), \, [[\Phi _{i_1}, \Phi _{i_2}], \Phi _{i_3}](z_0),\, \ldots \qquad (h, i_h\le k+1) \] belong to $T_{z_0}\mathbb{M}$. Such a property has been proved in [9] for $k=1$ and its proof in the case $k=2$ is the main purpose of the present paper. The following corollary follows at once: Let $\mathbb{D}$ be a $C^2$ distribution of rank $N$ on an open set $U\subset ^{N+m}$ and $\mathbb{M}$ be a $N$-dimensional $C^2$ submanifold of $U$. Moreover let $z_0\in \mathbb{M}$ be a $(N+2)$-density point of the tangency set $\lbrace z\in \mathbb{M}\,\vert \, T_z\mathbb{M}=\mathbb{D}(z)\rbrace $. Then $\mathbb{D}$ must be $2$-involutive at $z_0$, i.e., for every family $\lbrace X_j\rbrace _{j=1}^N$ of class $C^2$ in a neighborhood $V\subset U$ of $z_0$ which generates $\mathbb{D}$ one has \[ X_{i_1} (z_0), [X_{i_1},X_{i_2}](z_0), [[X_{i_1},X_{i_2}],X_{i_3}](z_0)\in T_{z_0}\mathbb{M}\] for all $1\le i_1, i_2, i_3\le N$.
References:
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