Symmetries of analytic curves
نویسندگان
چکیده
Analytic curves are classified w.r.t. their symmetry under a given regular and separately analytic Lie group action G×M→M on an manifold. We show that non-constant curve γ:D→M is either free or exponential – i.e., up to reparametrization of the form t↦exp(t⋅g→)⋅x. The vector g→∈g additionally proven be unique (non-zero scalation and) addition elements in algebra stabilizer Gγ≡{g∈G|g⋅γ=γ} γ. furthermore prove case, γ splits into countably many immersive subcurves each them discretely generated by G. This means such subcurve δ:D⊇(ι′,ι)→M build G-translates building block δ|Δ, whereby three different cases can occur: In shift blocks continuously distributed δ, with Δ always compact. Then, δ created iterated shifts δ|Δ some g∈G its inverse; class [e]≠[g]∈G/Gγ uniquely determined, as well same for possible decomposition. flip there exist defined compact interval, contained one only decomposition exists this case. Here, flips at boundary points these blocks, occurring transformations two non-trivial classes G/Gγ. mirror exactly (flipping) point δ(τ), translation [e]≠[g]∈G/Gγ. thus δ|(i′,τ], δ|[τ,i), δ|(i′,τ] flipped δ|[τ,i) vice versa (or both).
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ژورنال
عنوان ژورنال: Differential Geometry and Its Applications
سال: 2021
ISSN: ['1872-6984', '0926-2245']
DOI: https://doi.org/10.1016/j.difgeo.2020.101687