Abstract
We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.
Highlights
Differential invariants of various groups play an important role in applications [1,2,3].Classical curvatures of submanifolds in Euclidean space arise as differential invariants of the orthogonal group
Since the obtained invariants are quasi-linear in their respective top jet-variables, and this property is preserved by invariant derivations, the algebra A is generated by them
We computed the algebra of differential invariants for various geometric objects on symplectic spaces with several choices of the equivalence group and touched upon a relation between the invariants of the pair action
Summary
Differential invariants of various groups play an important role in applications [1,2,3]. In this paper we consider the linear symplectic group action and compute the corresponding algebra of differential invariants. We will concentrate on the linear case and compute the algebra of differential invariants for submanifolds and functions on V. Generators of the algebra of differential invariants will be presented in the Lie-Tresse form as functions and derivations, and for lower dimensions, we compute the differential syzygies. The Lie algebra method works well in dimension 2 (symplectic case n = 1) and fails further. We describe in turn differential invariants of functions, curves and hypersurfaces in symplectic vector spaces, and discuss the particular case of surfaces in. We briefly discuss the invariants in contact vector spaces and demonstrate how to compute differential invariants for conformal and affine extensions from our preceding computations. Some large formulae are delegated to the Appendix A, the other can be found as Supplementary Material in this article
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