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<p>In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb Z}}/2{{\mathbb Z}}\)</EquationSource> </InlineEquation>-structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896), and that the structure is locally isomorphic to a left-invariant one when this dimension is maximal. We also classify the invariant path structures on three-dimensional Lie groups.</p>

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Reductions of Path Structures and Classification of Homogeneous Structures in Dimension Three

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • E. Falbel,
  • Martin Mion-Mouton,
  • Jose Miguel Veloso

摘要

In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a \({{\mathbb Z}}/2{{\mathbb Z}}\) -structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896), and that the structure is locally isomorphic to a left-invariant one when this dimension is maximal. We also classify the invariant path structures on three-dimensional Lie groups.