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

Abstract

Given a principal bundle G \rightarrow P \rightarrow B (each being compact, connected and oriented) and a G-invariant metric h^{P} on P which induces a volume form \mu^{P}, we consider the group of all unimodular automorphisms SAut(P,\mu^{P}):={\varphi\in Diff(P) | \varphi^{*}\mu^{P}=\mu^{P} and \varphi is G-equivariant} of P and determines its Euler equation a 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,\mu^{P} is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P.