Differential invariants of immersions of manifolds with metric fields

The problem of finding all (higher order) differential invariants of immersions f : X → Y, where X and Y are manifolds endowed with metric fields, is investigated. The underlying jet manifolds are introduced, and the action of the corresponding differential groups on them are analysed. It is shown that the differential invariants can be described by means of the factorization of the group action with respect to a distinguished subgroup. By the orbit reduction method a basis of first-order invariants is found. In particular, this basis is used for a characterization of all first-order invariant Lagrangians depending on two metrics and an immersion.