Higher dimensional generalisation of the μ-Hunter-Saxton equation

Details Higher dimensional generalisation of the μ-Hunter-Saxton equation Higher dimensional generalisation of the μ-Hunter-Saxton equation

2012-03-20

arxiv.org/abs/1203.4463v1

Physics

Mathematical Physics

Scientific paper

A higher dimensional generalisation of the {\mu}-Hunter-Saxton ({\mu}HS) equation is studied. It is shown that this equation is the Euler-Arnold equation corresponding to geodesics in Diff(M) with respect to a right invariant metric. This is the first example of a right invariant non-degenerate metric on Diff(M) that descends properly to the space of densities Dens(M) = Diff(M)/Diffvol(M). The horizontal geodesics in Diff(M) descend to the recently considered \.H 1 geodesics in Dens(M). Furthermore, the horizontal geodesics originating from the identity element generates a local submanifold of Diff(M), which is locally diffeomorphic to the quotient space Dens(M) by a radial isometry. As an application, a result about (local) factorisation of Lie groups is given. Contrary to other factorisation results for Lie groups, such as the Cartan factorisation, this result is based on the Riemannian exponential rather than the group exponential.

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