Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Designing metrics; the delta metric for curves

created by mennucci on 12 Mar 2018
modified on 06 Feb 2023


Published Paper

Inserted: 12 mar 2018
Last Updated: 6 feb 2023

Journal: ESAIM: COCV
Volume: 25
Number: 59
Year: 2019
Doi: 10.1051/cocv/2018044


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 ``normalization''. 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.

Keywords: minimal geodesic, space of curves, Riemannian metric, invariant metric, group action, geodesic