We investigate some geometric properties of the curl operator, based on its diagonalization and its expression as a non-local symmetry of the pseudo-derivative (−∆)^{1/2} among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of (−∆)^{−1/2} curl. The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one.

The non-linearity of Navier-Stokes has the structure of a cross-product. In the case of a finite-time blow-up, both spin-definite components of the flow will explode simultaneously and with equal rates, i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows. We investigate the role of the local and non-local determinants det(curl(u),u,(−∆)^\theta u), which drive enstrophy and are responsible for the regularity of the flow or the emergence of singularities or quasi-singularities. As such, they are at the core of turbulence phenomena.