Conserved quantities of distinguished curves on conformal sphere
Prim Plansangkate, Lenka Zalabová
TL;DR
The paper develops two parallel routes to conserved quantities for distinguished curves on the conformal sphere: a tractor-calculus approach using parallel wedges of canonical tractors, and a Lagrangian/Noether approach for curves obeying the conformal Mercator equation. It shows that curves with a parallel 4-tractor, and those solving the conformal Mercator equation, admit explicit families of conserved quantities; crucially, there is a precise relation between these two sets of invariants established via a Hamiltonian formulation. The results yield explicit expressions for invariants in both spacetime and phase-space variables and include concrete examples such as logarithmic spirals and conformal circles, clarifying the overlap between the two generalizations. Overall, the work provides a unified framework connecting tractor-based and variationally derived conserved quantities for conformal-geometric curves on the conformal sphere, with potential applications to integrability and symmetry analysis in conformal geometry.
Abstract
We give conserved quantities of two generalizations of conformal circles on the conformal sphere. One generalization concerns curves carrying a distinguished parallel tractor along them, which can be used to construct the conserved quantities. The other is the class of curves satisfying the conformal Mercator equation, the conserved quantities of which are computed using Lagrangian formalism. The two generalizations are not disjoint, and our main result is the relation between their conserved quantities. Explicit examples are also presented.