On moving frames and Toda lattices of BKP and CKP types

Abstract

This paper is mainly concerned with the geometric formulations of Toda lattices of BKP and CKP types as evolutions of invariants and their explicit expressions. The theory for discrete curves in centro-affine geometry is constructed by using the method of a moving frame. With the help of orthogonal polynomial theory, we choose the appropriate evolutions for the curves and the evolutions induced on invariants are related to these Toda-type lattices. In addition, the explicit expressions for the discrete invariant curve flows, discrete moving frames and Maurer–Cartan invariants are provided.

Multi-component Toda lattice in centro-affine $${\mathbb R}^n$$

“…In this paper, we are going to apply the above theorem to the centro-affine space. In fact, equation (7) and condition (8) completely determine the evolution of K s [6,7,10]. Note that identity ( 7) is similar to the zero curvature condition (without the spectral parameter) for completely integrable systems.…”

“…This paper is devoted to the study of invariant evolutions in centro-affine R n and induced integrable systems. In 3-dimensional centro-affine case, the authors of [10] studied the geometric realisations of the B-Toda lattice and C-Toda. Recently Beffa and Calini investigated the evolutions of arc length-parametrised polygons (corresponding to the case p s = 1 in Section 4) in n-dimensional centro-affine space, which can be identified with the case of projective RP n−1 [11].…”