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Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds

Gloria Marí Beffa (2008)

Annales de l’institut Fourier

In this paper we describe moving frames and differential invariants for curves in two different $| 1 |$ -graded parabolic manifolds $G / H$ , $G = O (p + 1, q + 1)$ and $G = O (2 m, 2 m)$ , and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in $G / H$ inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in $G / H$ can be reduced...

Recursion in curve geometry.

Langer, Joel (1999)

The New York Journal of Mathematics [electronic only]

Surfaces in three-dimensional Lie groups.

Berdinskij, D.A., Tajmanov, I.A. (2005)

Sibirskij Matematicheskij Zhurnal

The geometry of dented pentagram maps

Boris Khesin, Fedor Soloviev (2016)

Journal of the European Mathematical Society

We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d - 1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...

The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in $ℂ P^{2}$ .

Mironov, A.E. (2004)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

The Spectrum of Jacobi Matrices.

Pierre van Moerbeke (1976)

Inventiones mathematicae