The geodesic motion on a Lie group equipped with a left or right invariant Riemannian
metric is governed by the Euler-Arnold equation. This paper investigates conditions on the
metric in order for a given subgroup to be totally geodesic. Results on the construction
and characterisation of such metrics are given. The setting works both in the classical nite
dimensional case, and in the category of in nite dimensional Fr echet Lie groups, in which
di eomorphism groups are included. Using the framework we give new examples of both nite
and in nite dimensional totally geodesic subgroups. In particular, based on the cross helicity,
we construct right invariant metrics such that a given subgroup of exact volume preserving
di eomorphisms is totally geodesic.
The paper also gives a general framework for the representation of Euler-Arnold equations
in arbitrary choice of dual pairing
