Discrete Moving Frames and Discrete Integrable Systems
Discrete Moving Frames and Discrete Integrable Systems
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DOI：
10.1007/s10208-013-9153-0
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发表时间：
2012-12
影响因子：
3
通讯作者：
E. Mansfield;G. M. Beffa;Jing Ping Wang
中科院分区：
文献类型：
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作者：
E. Mansfield;G. M. Beffa;Jing Ping Wang
基于群的运动框架具有广泛的应用，从微分几何中的经典等价问题到计算机视觉等更现代的应用。这里我们描述的是基于离散群的运动帧，它本质上是一个具有重叠域的运动帧序列。我们演示了不变量代数的一小组生成器，我们称之为离散Maurer-Cartan不变量，它们有递归公式。我们表明，这为我们研究离散可积系统提供了显著的计算优势。我们证明了一些曲率流的离散类似物自然地导致哈密顿对，从而产生可积的微分-差分系统。特别地，我们证明了在中心仿射平面和射影空间中，得到的哈密顿对可以在Miura变换下分别转化为Toda格和修正Volterra格的已知哈密顿对。我们还证明了中心仿射平面上多边形的一个特定不变映射可以转化为Toda晶格的可积离散化。此外，我们还详细地描述了均匀2球中的离散流，并得到了Volterra型方程作为多边形在球上的演化的实现。
Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call adiscrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.