Involutivity degree of a distribution at superdensity points of its tangencies

Delladio, Silvano. "Involutivity degree of a distribution at superdensity points of its tangencies." Archivum Mathematicum 057.4 (2021): 195-219. <http://eudml.org/doc/298062>.

@article{Delladio2021,
abstract = {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 $1\le i_1, i_2, i_3\le N$.},
author = {Delladio, Silvano},
journal = {Archivum Mathematicum},
keywords = {tangency set; distributions; superdensity; integral manifold; Frobenius theorem},
language = {eng},
number = {4},
pages = {195-219},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Involutivity degree of a distribution at superdensity points of its tangencies},
url = {http://eudml.org/doc/298062},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Delladio, Silvano
TI - Involutivity degree of a distribution at superdensity points of its tangencies
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 4
SP - 195
EP - 219
AB - 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 $1\le i_1, i_2, i_3\le N$.
LA - eng
KW - tangency set; distributions; superdensity; integral manifold; Frobenius theorem
UR - http://eudml.org/doc/298062
ER -