Almost-Riemannian geometry on Lie groups (Q2832097)

An almost Riemannian structure (ARS) \(g\) on an \(n\)-dimensional manifold is given by a set of \(n\) vector fields considered as an orthogonal frame of \(g\) outside the `singular' locus \({\mathcal Z}\) where they fail to be linear independent. Almost Riemannian structures have been studied by several authors in relation with control theory, see for instance papers on the two-dimensional case by A. Agrachev, U. Boscain and others cited in the article.NEWLINENEWLINEIn higher dimensions the situation is more involved, becoming simpler in the homogeneous case, especially when the manifold is a Lie group \(G\). Here, the authors study `simple ARSs' on \(G\), given by \(n\) vector fields \(\{\mathcal{X}, Y_1,\ldots,Y_{n-1}\}\) ones where the vector field \(\mathcal{X}\) is linear, i.e., its flow is a one-parameter group of automorphisms. The remaining \(n-1\), \(n=\dim(G)\), are left-invariant. A simple abelian example is given by \(G=\mathbb{R}^n\), each \(Y_i\) a constant vector and \(\mathcal{X}\) is represented by an \(n\times n\) matrix. A simple ARS on an \(n\)-dimensional Lie group \(G\) is related to a codimension-one regular sub-Riemannian structure on the \((n+1)\)-dimensional semidirect product \(\widetilde{G}=G\rtimes_\varphi\mathbb{R}\), where \(\varphi:\mathbb{R}\to\mathrm{Aut}(G)\) is the 1-parameter group of automorphisms generated by \(\mathcal{X}\)NEWLINENEWLINEAfter stating some basic definitions and facts, sufficient conditions for \({\mathcal Z}\) to be a submanifold or a subgroup are stated. These are given in terms of algebraic properties of the subspace \(\Delta\) generated by \(Y_1,\ldots,Y_{n-1}\) of the Lie algebra \(\mathfrak{g}\) of \(G\) and the derivation induced on the Lie algebra \(\mathfrak{g}\) OF \(g\) by \(\mathcal{X}\).NEWLINENEWLINEThe study is illustrated by examples, in particular the Heisenberg group, the group of affine motions of the real line and the special linear group \(\mathrm{SL}(2;\mathbb{ R })\). These show that the ARSs on Lie groups, even the simple ones, go much further than the two-dimensional (2D) or 3D generic case. The Hamiltonian equations of the Pontryagin maximum principle applied to simple ARSs on \(G\) are given.NEWLINENEWLINEAt the end of the article these considerations are generalized to homogeneous spaces and to the case of \(n\) affine vector fields instead of \(n-1\) left-invariant \(Y_i\) and one linear field \(\mathcal{X}\).