Quaternionic Geometry of Chiral Helices on a Cone: Riemannian Structure, Topological Classification, Variational Characterization,and Holomorphic Representation

Using quaternions as the algebraic tool, this paper systematically constructs the Riemannian geometric theory of chiral helices on a cone. The cone C_α( u) is determined by a unit purely imaginary quaternion u and the semi-vertical angle α; its induced metric is ds² = dv² + v²sin²α\,dφ², its Gaussian curvature vanishes identically, and it can be isometrically unfolded onto a planar sector. Chiral helices are generated by the one-parameter group of quaternionic screw motions Φₜ( v) = e-λ teuω t/2 v e- uω t/2, with parametric equations v(t)=v₀e-λ t, φ(t)=θ₀+ω t, and the chirality parameter is defined as σ=sgn(ω). The central result of this paper is that **the sign of the geodesic curvature is exactly the chirality parameter**, i.e., sgn(kg)=σ. Around this core result, the chiral helix is systematically characterized from five perspectives — differential geometry, topology, variational calculus, complex analysis, and dynamical systems: - **Differential geometry**: explicit formulas are derived for the Frenet curvature κ, the geodesic curvature kg, the normal curvature kₙ, the geodesic torsion τg, and the Frenet torsion τ, and it is proved that τ=τg (since kₙ/kg is constant); - **Topology**: radially closed helices ΓT are constructed, and it is proved that the linking number satisfies Lk(ΓT)=Nsgn(ω); mirror reflection reverses chirality, and a closed helix is not orientation-preservingly isotopic to its mirror image; - **Variational calculus**: it is proved that the helix is not a stationary point of the standard bending energy, but is a critical curve of the weighted bending energy Eweight=∫ v\,kg²\,ds; this functional is invariant under the radial rescaling v↦ av, and its family of critical curves is precisely the class of all logarithmic spirals; - **Complex analysis**: complex coordinates z=veiφ on the cone are introduced, yielding the holomorphic representation z(t)=z₀eᶜᵗ of the helix (c=-λ+iωsinα); in logarithmic coordinates w=log z the helix becomes a straight line; - **Dynamical systems**: the screw motion is regarded as a continuous dynamical system on the cone; it is proved that the apex is a global attractor, both Lyapunov exponents are equal to -λ, no nontrivial finite invariant measure exists, and the section map is a linear contraction P(v)=ρ v (ρ=e-2πλ/|ω|). The paper further points out that criticality with respect to the weighted bending energy is not exclusive to the helix but is a property shared by all logarithmic spirals; the helix is the representative among them satisfying the parameter relation a=-λ/(ωsinα). Higher-dimensional generalizations (higher-dimensional cones, helices in quaternionic projective space) and several open problems are also discussed.