Integrability conditions on Engel-type manifolds

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 $$Z_1 f = a_1$$ , $$ Z_2 f = a_2$$ . These are given by $$ Z_1^2 a_2 = (Z_1 Z_2 +[Z_1, Z_2]) a_1$$ , $$ Z_2^3 a_1 = (Z_2^2 Z_1 - Z_2 [Z_1, Z_2] - [Z_2, [Z_1, Z_2] ]) a_2$$ . Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincare-type lemma for the Engel’s distribution.