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.
Analysis and Mathematical Physics – Springer Journals
Published: Jun 23, 2015
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