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DESIGNING METRICS; THE DELTA METRIC FOR CURVES

机译：设计指标;曲线的DELTA度量

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摘要

In the first part, we revisit some key notions. Let M be a Riemannian manifold. Let G be a group acting on M. We discuss the relationship between the quotient M/G, "horizontality" and ormalization". We discuss the distinction between path-wise invariance and point-wise invariance and how the former positively impacts the design of metrics, in particular for the mathematical and numerical treatment of geodesics. We then discuss a strategy to design metrics with desired properties.In the second part, we prepare methods to normalize some standard group actions on the curve; we design a simple differential operator, called the delta operator, and compare it to the usual differential operators used in defining Riemannian metrics for curves.In the third part we design two examples of Riemannian metrics in the space of planar curves. These metrics are based on the delta" operator; they are "modular", they are composed of different terms, each associated to a group action. These are "strong" metrics, that is, smooth metrics on the space of curves, that is defined as a differentiable manifolds, modeled on the standard Sobolev space H-2. These metrics enjoy many important properties, including: metric completeness, geodesic completeness, existence of minimal length geodesics. These metrics properly project on the space of curves up to parameterization; the quotient space again enjoys the above properties.
机译：在第一部分中，我们重新审视一些关键概念。令M为黎曼流形。令G为作用于M的组。我们讨论商M / G，“水平”和“归一化”之间的关系。我们讨论路径不变性和点不变性之间的区别，以及前者如何对设计产生积极影响度量，尤其是用于大地测量学的数学和数值处理，然后讨论一种设计具有所需属性的度量的策略。第三部分，我们在平面曲线的空间中设计了两个黎曼度量的示例。这些度量基于 delta算符;它们是“模块化”的，它们由不同的术语组成，每个术语都与一个组动作相关联。这些是“强”度量，即在标准Sobolev空间H-2上建模的曲线空间（定义为可微流形）上的平滑度量。这些度量标准具有许多重要属性，包括：度量标准完整性，测地线完整性，最小长度测地线的存在。这些度量正确地投影在曲线的空间上，直到参数化为止。商空间再次具有上述特性。

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