Are there two smooth independent vector fields $X,Y$ on $S^3$ with $[X,Y]=Y$?



Independent vector fields $X,Y$ on $S^3$ with $[X,Y]=Y$
No. If such a pair existed, they would be tangent to a codimension $1$ foliation of $S^3$. Such a foliation must have a Reeb component, in particular, a compact leaf, $T\subset S^3$, which would be a torus (since its tangent bundle would be trivial). If $\xi$ and $\eta$ were the $1$-forms on $T$ dual to the basis $X$ and $Y$ of its tangent bundle, then the above equation would imply that $\mathrm{d}\eta = \eta\wedge\xi$, but this would make a nonvanishing $2$-form on $T$ have vanishing cohomology class in $H^2(T,\mathbb{R})$ (de Rham cohomology), which is clearly impossible.


