| Title:
|
Involutivity degree of a distribution at superdensity points of its tangencies (English) |
| Author:
|
Delladio, Silvano |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
57 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
195-219 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $1\le i_1, i_2, i_3\le N$. (English) |
| Keyword:
|
tangency set |
| Keyword:
|
distributions |
| Keyword:
|
superdensity |
| Keyword:
|
integral manifold |
| Keyword:
|
Frobenius theorem |
| MSC:
|
28Axx |
| MSC:
|
58A17 |
| MSC:
|
58A30 |
| MSC:
|
58C35 |
| idZBL:
|
Zbl 07442412 |
| idMR:
|
MR4346111 |
| DOI:
|
10.5817/AM2021-4-195 |
| . |
| Date available:
|
2021-10-06T08:53:37Z |
| Last updated:
|
2022-02-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149129 |
| . |
| Reference:
|
[1] Balogh, Z.M.: Size of characteristic sets and functions with prescribed gradient.J. Reine Angew. Math. 564 (2003), 63–83. MR 2021034 |
| Reference:
|
[2] Balogh, Z.M., Pintea, C., Rohner, H.: Size of tangencies to non-involutive distributions.Indiana Univ. Math. J. 60 (6) (2011), 2061–2092. MR 3008261, 10.1512/iumj.2011.60.4489 |
| Reference:
|
[3] Chavel, I.: Riemannian Geometry: A Modern Introduction.Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, 1995. |
| Reference:
|
[4] Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on differential geometry.Series On University Mathematics, vol. 1, World Scientific, 1999. |
| Reference:
|
[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. MR 3683451, 10.1002/mana.201600195 |
| Reference:
|
[6] Delladio, S.: Structure of tangencies to distributions via the implicit function theorem.Rev. Mat. Iberoam. 34 (3) (2018), 1387–1400. MR 3850291, 10.4171/RMI/1028 |
| Reference:
|
[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. MR 3954388, 10.1007/s10231-018-0793-1 |
| Reference:
|
[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. MR 3951069, 10.1512/iumj.2019.68.7549 |
| Reference:
|
[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. MR 4218579, 10.1007/s10476-020-0063-5 |
| Reference:
|
[10] Derridj, M.: Sur un théorème de traces.Ann. Inst. Fourier (Grenoble) 22 (2) (1972), 73–83. 10.5802/aif.413 |
| Reference:
|
[11] Federer, H.: Geometric Measure Theory.Springer-Verlag, 1969. Zbl 0176.00801 |
| Reference:
|
[12] Narasimhan, R.: Analysis on real and complex manifolds.North-Holland Math. Library, North-Holland, 1985. |
| . |
