Geometric Structure of Vector Fields on Surfaces—Toward a Nonlocal Definition of Vortex

The notion of a vortex is fundamental in fluid dynamics, where it broadly refers to rotary fluid motions of various forms. Yet a precise, universal definition has remained elusive. Two fundamental challenges persist in the existing definitions of a vortex. The first is the gap between the analytic perspective that adopts a framework of motion decomposition and the synthetic perspective that emphasizes the geometric patterns of the composite motion. The second is the gap between precisely defined local measures of rotation and intuitive large-scale descriptions of vortices. This paper develops a geometric theory of smooth tangent vector fields on oriented closed surfaces that bridges the analytic and synthetic perspectives, and provides a nonlocal definition of a vortex core. Working within the frameworks of the irreducible symmetric-antisymmetric decomposition (iSAD), eigenvalue decomposition (EVD) and Helmholtz-Hodge decomposition (HHD), we prove two principal results for such vortex cores. First, all streamlines therein wind in the same direction, indicating a nonlocal rotary motion in the entire vortex core. Second, each vortex core can have at most one axis (center or focus). The theory is illustrated with examples on spheres and tori of various curvature, demonstrating how geometry and topology shape the shape of vortex cores. The results are purely mathematical and extend naturally to open surfaces, offering a rigorous foundation for vortex identification across disciplines.