Randomized trial explores global helical vector fields on the three-sphere, indicating unique geometrical foundations.
Key Points
This work aims to establish a comprehensive geometric theory of helical vector fields on the three-sphere, integrating key mathematical frameworks.
Developed the theory based on non-singular Riemannian geometry and $SO(4)$ isometry decomposition.
Explored the spectral characteristics using the Laplace–Beltrami operator and the spectral entropy functional.
Extended the phase diagram to include helical strength as a control parameter.
Established a unique globally smooth helical vector field on $S^3$ with an orthogonal decomposition.
Derived the correspondence between the helical strength parameter $ ext{(kappa)}$ and critical scale $a_{ ext{crit}}(k)$.
Demonstrated that the spectral entropy $S_{ ext{spec}}$ evolves monotonically with respect to $ ext{(kappa)}$ below a unique threshold $ ext{(kappa}_{ ext{crit}})$, indicating changes in geometric stability.