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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {g}_0< \mathfrak {g}_1< \mathfrak {g}_2< \dots< \mathfrak {g}_{n-1}<\mathfrak {g}_n=\mathfrak {g}=Lie(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">g</mi> <mn>0</mn> </msub> <mo><</mo> <msub> <mi mathvariant="fraktur">g</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi mathvariant="fraktur">g</mi> <mn>2</mn> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi mathvariant="fraktur">g</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi mathvariant="fraktur">g</mi> <mi>n</mi> </msub> <mo>=</mo> <mi mathvariant="fraktur">g</mi> <mo>=</mo> <mi>L</mi> <mi>i</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In the case where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathfrak g_0=Lie(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">g</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>L</mi> <mi>i</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Normal Sub-Riemannian Geodesics Related to Filtrations of Lie Algebras

期刊论文
发表日期：

2026年5月11日

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Božidar Jovanović,
Tijana Šukilović,
Srdjan Vukmirović

摘要

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 $\mathfrak {g}_0< \mathfrak {g}_1< \mathfrak {g}_2< \dots< \mathfrak {g}_{n-1}<\mathfrak {g}_n=\mathfrak {g}=Lie(G)$ $g_{0} < g_{1} < g_{2} < \dots < g_{n - 1} < g_{n} = g = L i e (G)$ . In the case where $n=1$ $n = 1$ , 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 $\mathfrak g_0=Lie(K)$ $g_{0} = L i e (K)$ .