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 $C_\alpha(\mathbf u)$ is determined by a unit purely imaginary quaternion $\mathbf u$ and the semi-vertical angle $\alpha$; its induced metric is $ds^2 = dv^2 + v^2\sin^2\alpha\,d\varphi^2$, 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 $\Phi_t(\mathbf v) = e^{-\lambda t}e^{\mathbf u\omega t/2}\mathbf v e^{-\mathbf u\omega t/2}$, with parametric equations $v(t)=v_0e^{-\lambda t}$, $\varphi(t)=\theta_0+\omega t$, and the chirality parameter is defined as $\sigma=\operatorname{sgn}(\omega)$. The central result of this paper is that **the sign of the geodesic curvature is exactly the chirality parameter**, i.e., $\operatorname{sgn}(k_g)=\sigma$. 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 $\kappa$, the geodesic curvature $k_g$, the normal curvature $k_n$, the geodesic torsion $\tau_g$, and the Frenet torsion $\tau$, and it is proved that $\tau=\tau_g$ (since $k_n/k_g$ is constant); - **Topology**: radially closed helices $\Gamma_T$ are constructed, and it is proved that the linking number satisfies $\operatorname{Lk}(\Gamma_T)=N\operatorname{sgn}(\omega)$; 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 $E_{\text{weight}}=\int v\,k_g^2\,ds$; this functional is invariant under the radial rescaling $v\mapsto av$, and its family of critical curves is precisely the class of all logarithmic spirals; - **Complex analysis**: complex coordinates $z=ve^{i\phi}$ on the cone are introduced, yielding the holomorphic representation $z(t)=z_0e^{ct}$ of the helix ($c=-\lambda+i\omega\sin\alpha$); 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 $-\lambda$, no nontrivial finite invariant measure exists, and the section map is a linear contraction $P(v)=\rho v$ ($\rho=e^{-2\pi\lambda/|\omega|}$). 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 $a=-\lambda/(\omega\sin\alpha)$. Higher-dimensional generalizations (higher-dimensional cones, helices in quaternionic projective space) and several open problems are also discussed.