Geodesic
Involutivity degree of a distribution at superdensity points of its tangencies
Archivum mathematicum, Tome 57 (2021) no. 4, pp. 195-219
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $1\le i_1, i_2, i_3\le N$.
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 $1\le i_1, i_2, i_3\le N$.
DOI : 10.5817/AM2021-4-195
Classification : 28Axx, 58A17, 58A30, 58C35
Keywords: tangency set; distributions; superdensity; integral manifold; Frobenius theorem 
@article{10_5817_AM2021_4_195,
     author = {Delladio, Silvano},
     title = {Involutivity degree of a distribution at superdensity points of its tangencies},
     journal = {Archivum mathematicum},
     pages = {195--219},
     year = {2021},
     volume = {57},
     number = {4},
     doi = {10.5817/AM2021-4-195},
     mrnumber = {4346111},
     zbl = {07442412},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2021-4-195/}
}
TY  - JOUR
AU  - Delladio, Silvano
TI  - Involutivity degree of a distribution at superdensity points of its tangencies
JO  - Archivum mathematicum
PY  - 2021
SP  - 195
EP  - 219
VL  - 57
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2021-4-195/
DO  - 10.5817/AM2021-4-195
LA  - en
ID  - 10_5817_AM2021_4_195
ER  - 
%0 Journal Article
%A Delladio, Silvano
%T Involutivity degree of a distribution at superdensity points of its tangencies
%J Archivum mathematicum
%D 2021
%P 195-219
%V 57
%N 4
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2021-4-195/
%R 10.5817/AM2021-4-195
%G en
%F 10_5817_AM2021_4_195
Delladio, Silvano. Involutivity degree of a distribution at superdensity points of its tangencies. Archivum mathematicum, Tome 57 (2021) no. 4, pp. 195-219. doi: 10.5817/AM2021-4-195

[1] Balogh, Z.M.: Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564 (2003), 63–83. | MR

[2] Balogh, Z.M., Pintea, C., Rohner, H.: Size of tangencies to non-involutive distributions. Indiana Univ. Math. J. 60 (6) (2011), 2061–2092. | DOI | MR

[3] Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, 1995.

[4] Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on differential geometry. Series On University Mathematics, vol. 1, World Scientific, 1999.

[5] Delladio, S.: A note on a generalization of the Schwarz theorem about the equality of mixed partial derivatives. Math. Nachr. 290 (11–12) (2017), 1630–1636, DOI: 10.1002/mana.201600195. | DOI | MR

[6] Delladio, S.: Structure of tangencies to distributions via the implicit function theorem. Rev. Mat. Iberoam. 34 (3) (2018), 1387–1400. | DOI | MR

[7] Delladio, S.: Structure of prescribed gradient domains for non-integrable vector fields. Ann. Mat. Pura Appl. 198 (3) (2019), 685–691, DOI: 10.1007/s10231-018-0793-1. | DOI | MR

[8] Delladio, S.: The tangency of a $C^1$ smooth submanifold with respect to a non-involutive $C^1$ distribution has no superdensity points. Indiana Univ. Math. J. 68 (2) (2019), 393–412. | DOI | MR

[9] Delladio, S.: Good behaviour of Lie bracket at a superdensity point of the tangency set of a submanifold with respect to a rank $2$ distribution. Anal. Math. 47 (1) (2021), 67–80. | DOI | MR

[10] Derridj, M.: Sur un théorème de traces. Ann. Inst. Fourier (Grenoble) 22 (2) (1972), 73–83. | DOI

[11] Federer, H.: Geometric Measure Theory. Springer-Verlag, 1969. | Zbl

[12] Narasimhan, R.: Analysis on real and complex manifolds. North-Holland Math. Library, North-Holland, 1985.

Cité par Sources :