Isotropic curve flows on R^{n+1, n}

2016/08/26 by Chuu-Lian Terng, Zhiwei Wu, Terng, Chuu-Lian +1
Mathematics · Physics and Astronomy · #37K05 #37K10 #37K25 #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1608.07628

openalex publication_date 2016/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R^{n+1, n} be the vector space R²ⁿ⁺¹ equipped with the bilinear form (X,Y)=X^t C_n Y of index n, where C_n= ∑_i=1²ⁿ⁺¹ (-1)^n+i-1 e_{i, 2n+2-i}. A smooth γ: R→ R^n+1,n is \it isotropic if γ, γ_x, …, γ_x⁽²ⁿ⁾ are linearly independent and the span of γ, …, γ_x^(n-1) is isotropic. Given an isotropic curve, we show that there is a unique up to translation parameter such that (γ_x⁽ⁿ⁾, γ_x⁽ⁿ⁾)=1 (we call such parameter the isotropic parameter) and there also exists a natural moving frame. In this paper, we consider two sequences of curve flows on the space of isotropic curves parametrized by isotropic parameter. We show that differential invariants of these isotropic curves satisfy Drinfeld-Sokolov's KdV type soliton hierarchies associated to the affine Kac-Moody algebra B_n⁽¹⁾ and A_2n⁽²⁾ Then we use techniques from soliton theory to construct bi-Hamiltonian structure, conservation laws, Backlund transformations and permutability formulas for these curve flows.

Citations

• N-dimension Central Affine Curve Flows
• On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ

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