Local Symplectic Invariants for Curves

To the best of our knowledge, the study of the local symplectic invariants of submanifolds of Euclidean space was initiated by Chern and Wang in 1947, [6]. They considered mainly the case of curves and hypersurfaces, and obtained structure equations defining a set of local symplectic differential invariants for these objects. We should explain at this stage that by “symplectic invariants” we mean invariants under the direct product of the affine linear symplectic group of R endowed with the standard symplectic form with the infinite-dimensional pseudo-group of reparametrizations of the submanifolds. This is in contrast with the case in which one considers the full infinite-dimensional symplectomorphism group of the ambient R. Indeed, in the latter case, the theorem of Darboux implies that submanifolds have no local differential invariants. (On the other hand, Ekeland and Hofer’s discovery of symplectic capacity, [7], shows that there are nontrivial global invariants. These lie beyond the scope of this work.). The case we are interested in is much closer in spirit to ordinary Euclidean or affine