Integrability conditions on a sub-Riemannian structure on $\mathbb {S}^3$

Abstract

This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere $\mathbb {S}^3$. We prove that a certain sub-Riemannian system $Xf =a$, $Yf =b$ on $\mathbb {S}^3$ has a solution if and only if the following integrability conditions hold: $X^2 b + 4b = (XY + [X, Y]) a$, $Y^2 a + 4a = (YX-[X, Y]) b$. We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b.