  Lagrangian averaging theories, most notably the Generalised Lagrangian Mean
(GLM) theory of Andrews & McIntyre (1978), have been primarily developed in
Euclidean space and Cartesian coordinates. We re-interpret these theories using
a geometric, coordinate-free formulation. This gives central roles to the flow
map, its decomposition into mean and perturbation maps, and the momentum 1-form
dual to the velocity vector. In this interpretation, the Lagrangian mean of any
tensorial quantity is obtained by averaging its pull back to the mean
configuration. Crucially, the mean velocity is not a Lagrangian mean in this
sense. It can be defined in a variety of ways, leading to alternative
Lagrangian mean formulations that include GLM and Soward & Roberts' (2010) glm.
These formulations share key features which the geometric approach uncovers. We
derive governing equations both for the mean flow and for wave activities
constraining the dynamics of the pertubations. The presentation focusses on the
Boussinesq model for inviscid rotating stratified flows and reviews the
necessary tools of differential geometry.
