Analytical Properties of Sobolev Mappings on Roto-Translation Groups

The roto-translation group, $SE(2)$, is a three-dimensional topological manifold diffeomorphic to $\mathbb{R}^{2}\times\mathbb{S}^{1}$ with coordinates $(x,y,\theta)$. The left-invariant vector fields
$$ X_{1}=\cos\theta\frac{\partial}{\partial x}+\sin\theta\frac{\partial}{\partial y},\; X_{2}=\frac{\partial}{\partial\theta},\; X_{3}=-\sin\theta\frac{\partial}{\partial x}+\cos\theta\frac{\partial}{\partial y}, $$
form a basis of the Lie algebra of $SE(2)$. The bracket-generating subbundle of the tangent-bundle is spanned by the frame $X_{1}$, $X_{2}$.
Consider the basic 1-forms $dX_{1}$, $dX_{2}$, $dX_{3}$, dual to the basic vector fields $X_{1}$, $X_{2}$, $X_{3}$, i. e., $dX_{i}(X_{j})=\delta_{ij}$. Applying the methods developed in [1] we establish a key relation underlying the connection between mappings with bounded distortion [2] and nonlinear potential theory.
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Theorem. Let $SE(2)$ be a roto-translation group and $\Omega\subset SE(2)$ is an open set. Suppose that $f\colon\Omega\to SE(2)$ is a Sobolev mapping of the class $W^{1}_{4,\mathrm{loc}}(\Omega)$, $V\colon SE(2)\to\mathbb{R}^2$ is a vector field $V=(v_{1},v_{2})\in C^{1}$ such that $\mathrm{div}_{h}V=X_{1}v_{1}+X_{2}v_{2}$ is bounded on $SE(2)$, and
$$ \omega(g)=v_{1}(g)\,dX_{2}\wedge dX_{3}-v_{2}(g)\,dX_{1}\wedge dX_{3},\quad g\in\Omega. $$
Then the equality $df^{\#}\omega=f^{\#}d\omega$ holds in the sense of distributions.