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A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

Radu Slobodeanu

Abstract

We present a steady Euler flow on the round 3-sphere whose velocity vector field has the property of having two independent first integrals, being tangent to the fibres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potential.

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

Abstract

We present a steady Euler flow on the round 3-sphere whose velocity vector field has the property of having two independent first integrals, being tangent to the fibres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potential.

Paper Structure

This paper contains 7 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. $S$-integrable Euler fields and Faddeev-Skyrme solutions
  3. A new solution
  4. Appendix
  5. Cartesian coordinates.
  6. Hopf coordinates.
  7. Spherical coordinates.

Key Result

Proposition 1

Let $\varphi=\pi \circ \psi : \mathbb{S}^3 \to \mathbb{S}^2(\tfrac{1}{2})$ be the mapping defined above, and consider the spheres endowed with the usual round metrics (with $\omega$ the associated area 2-form on the codomain). Then $(i)$ the vector field $V=(\ast \varphi^* \omega)^\sharp$ is an $\m $(ii)$ the almost subersion $\varphi$ is a smooth critical point of Hopf invariant $Q(\varphi)=2$

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof