The group of unimodular automorphisms of a principal bundle and the Euler–Yang–Mills equations

Abstract

Given a principal bundle G ↪ P → B (each being compact, connected and oriented) and a G-invariant metric h P on P which induces a volume form μ P , we consider the group of all unimodular automorphisms SAut ( P , μ P ) : = { φ ∈ Diff ( P ) | φ ∗ μ P = μ P and φ is G -equivariant } of P, and determines its Euler equation à la Arnold. The resulting equations turn out to be (a particular case of) the Euler–Yang–Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut ( P , μ P ) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau ( P ) of P.