Normal sub-Riemannian geodesics related to filtrations of Lie algebras

TL;DR

The paper develops a general framework for left-invariant sub-Riemannian geodesics on Lie groups using filtrations of Lie algebras, extending the Agrachev–Brockett–Jurdjevic formula to chains of subalgebras and establishing complete non-commutative integrability. It provides explicit reconstruction formulas for normal geodesics, analyzes reduction to homogeneous spaces, and illustrates the theory with detailed examples on SU(3)<G2<SO(7), U(1)<SU(2)<U(2)<SO(4), and SO(n) bracket-generating distributions. Connections to Gel'fand–Cetlin integrable systems and to sub-Riemannian Manakov metrics are drawn, showing how integrals from these structures govern geodesic flows and their reductions. The work advances explicit, integrable sub-Riemannian geodesic flows on Lie groups and their homogeneous spaces, with implications for geometric control and the study of symmetric structures in differential geometry.

Abstract

There is a natural way to construct sub-Riemannian structures that depend on $n$ parameters on compact Lie groups. These structures are related to the filtrations of Lie subalgebras $\mathfrak g_0 < \mathfrak g_1 < \mathfrak g_2 < \dots < \mathfrak g_{n-1}<\mathfrak g_n=\mathfrak g=Lie(G)$. In the case where $n=1$, the explicit solution for normal sub-Riemannian geodesics was provided by Agrachev, Brockett, and Jurjdevic. We extend their solution to apply to general chains of Lie subgroups. Additionally, we describe normal geodesic lines of the induced sub-Riemannian structures on homogeneous spaces $G/K$, where $\mathfrak g_0=Lie(K)$.

Figures (3)

• Figure 1: Projection of the solution curves of the Euler equation on the subspace $\mathop{\mathrm{\mathrm{span\,}}}\nolimits\{\mathbf{e}_{13},\mathbf{e}_{14},\mathbf{e}_{24}\}$ for the distributions associated with $\mathfrak d=\mathfrak p_1\oplus\mathfrak p_3$ (left), and $\mathfrak d=\mathfrak p_1\oplus\mathfrak p_{2}\oplus\mathfrak p_3$ (right), with the same initial conditions.
• Figure 2: Projection of the integral curve of the system \ref{['novi']} on the subspace $\mathfrak{so}(3)=\mathop{\mathrm{\mathrm{span\,}}}\nolimits\{\mathbf{e}_{23},\mathbf{e}_{24},\mathbf{e}_{34}\}$ and the level set $I_2=-\frac{1}{2}$ (left), and $I_2=0$ (right). In the later case, the integral curve belongs to $\mathfrak{so}(3)$.
• Figure 3: Integral curves for the pair $(\mathfrak{so}(4),\mathfrak{so}(2)\otimes \mathfrak{so}(2))$ and Manakov metrics with $a_1=a_2$, $a_3=a_4$ (left), and $a_1\neq a_2$, $a_3\neq a_4$ (right).

Theorems & Definitions (22)

• Theorem 1
• Example 1
• proof
• Theorem 2
• proof
• Remark 1
• Example 2
• Theorem 3
• Remark 2
• Lemma 4