Minus one Homogeneous Euler Flows are Geodesible

Abstract

In this paper, we study $(-1)$-homogeneous steady solutions to the Euler equations on ${\mathbb{R}}^n\setminus \{0\}$. In low dimensions $n=2,3$, such flows are known to be essentially trivial. In contrast, we show that in higher dimensions $n\ge 4$, every $(-1)$-homogeneous Euler flow is a geodesible vector field with constant Bernoulli function. Moreover, any $(-1)$-homogeneous geodesible field is induced by a geodesible field on the sphere ${\mathbb{S}}^{n-1}$. In particular, in the case $n=4$, every $(-1)$-homogeneous Euler flow is obtained as an extension of a Beltrami field on ${\mathbb{S}}^3$.

19 pages