# Integrability conditions on Engel-type manifolds

Integrability conditions on Engel-type manifolds We introduce the concept of Engel manifold, as a manifold that resembles locally the Engel group, and find the integrability conditions of the associated sub-elliptic system \$\$Z_1 f = a_1\$\$ Z 1 f = a 1 , \$\$ Z_2 f = a_2\$\$ Z 2 f = a 2 . These are given by \$\$ Z_1^2 a_2 = (Z_1 Z_2 +[Z_1, Z_2]) a_1\$\$ Z 1 2 a 2 = ( Z 1 Z 2 + [ Z 1 , Z 2 ] ) a 1 , \$\$ Z_2^3 a_1 = (Z_2^2 Z_1 - Z_2 [Z_1, Z_2] - [Z_2, [Z_1, Z_2] ]) a_2\$\$ Z 2 3 a 1 = ( Z 2 2 Z 1 - Z 2 [ Z 1 , Z 2 ] - [ Z 2 , [ Z 1 , Z 2 ] ] ) a 2 . Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincaré-type lemma for the Engel’s distribution. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Integrability conditions on Engel-type manifolds

, Volume 5 (3) – Jun 23, 2015
15 pages

/lp/springer-journals/integrability-conditions-on-engel-type-manifolds-cNGF8E02TX
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-015-0107-3
Publisher site
See Article on Publisher Site

### Abstract

We introduce the concept of Engel manifold, as a manifold that resembles locally the Engel group, and find the integrability conditions of the associated sub-elliptic system \$\$Z_1 f = a_1\$\$ Z 1 f = a 1 , \$\$ Z_2 f = a_2\$\$ Z 2 f = a 2 . These are given by \$\$ Z_1^2 a_2 = (Z_1 Z_2 +[Z_1, Z_2]) a_1\$\$ Z 1 2 a 2 = ( Z 1 Z 2 + [ Z 1 , Z 2 ] ) a 1 , \$\$ Z_2^3 a_1 = (Z_2^2 Z_1 - Z_2 [Z_1, Z_2] - [Z_2, [Z_1, Z_2] ]) a_2\$\$ Z 2 3 a 1 = ( Z 2 2 Z 1 - Z 2 [ Z 1 , Z 2 ] - [ Z 2 , [ Z 1 , Z 2 ] ] ) a 2 . Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincaré-type lemma for the Engel’s distribution.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jun 23, 2015