Discrete moving frames on lattice varieties and lattice-based multispaces
The aim of this paper is to extend the theory of the discrete moving frame in two different ways.The first is to consider a discrete moving frame defined on a lattice variety, which can be thought of as the vertices, or 0-cells, together with their adjacency information, in a discrete approximation of a manifold. The authors describe their associated cross sections and define Maurer-Cartan invariants and local syzygies. They describe the equivalence classes of global syzygies that result from the first fundamental group of the variety. The authors consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, they construct multispace, a generalisation of the jet bundle that also generalises Olver's one-dimensional construction. Using interpolation to provide coordinates, they prove that it is a manifold containing the usual jet bundle as a submanifold and show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. The authors prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In their last chapter the authors discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, they discuss a more general result concerning the discretisation of smooth moving frames, and the continuum limit of equicontinuous families of discrete moving frames, with an example. More precisely, they use the Arzela-Ascoli theorem to give a general convergence result for an equicontinuous family of moving frames. This provides a rigorous foundation to a variety of examples involving the discretisation of a smooth frame.
discrete and smooth Maurer-Cartan invariants
finite difference calculus of variations
scientific article; zbMATH DE number 6849392
scientific article