Loxodromic unit vector field on punctured spheres

TL;DR

This paper characterizes loxodromic unit vector fields on punctured spheres, identifying those that minimize volume based on Poincaré indexes at singularities, thus linking geometric structures with topological invariants.

Contribution

It provides a unique characterization of loxodromic vector fields on punctured spheres as volume minimizers tied to Poincaré indexes, a novel geometric-topological link.

Findings

Loxodromic vector fields are the only volume minimizers on punctured spheres.

Volume bounds depend explicitly on Poincaré indexes.

The work links geometric properties with topological invariants.

Abstract

In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincar\'e indexes around their singularities.