Metrics of curves in shape optimization and analysis - Andrea Carlo ...

8.2.2 RepresentationThe Stiefel manifold is a complete smooth Riemannian manifold; it contains the(representation of) all closed rectifiable parametric planar curves, up to scalingand translation. So with this choice of metric and representation, we obtainthat the completion of the Fréchet manifold of smooth curves is a Riemanniansmooth manifold. There are two problems left.• The quotient w.r.to reparameterization. This is studied in [39], where itis proven that unfortunately geodesics in the quotient space may developsingularities in the reparameterizations at the end times of the geodesic.• But there is also the quotient w.r.to representation.8.2.3 Quotient w.r.to representationAs in the previous section, we may define a measurable square root lifting; thislifting will not be unique. Indeed, let (e, f) be the square root representation ofξ, that is ξ ′ = (e + if) 2 ; choose any function a : S 1 → {−1, 1} arbitrarily (butmeasurable); then (ae, af) represents the same curve. So again we may define aquotient metric ˆd(ξ 1 , ξ 2 ) as we did in eqn. (8.3); similar comments to those atthe end of Section 8.1.2 hold.9 Riemannian metrics of immersed curvesMetrics of “geometric” curves have been studied by Michor and Mumford [37, 36,38] and Yezzi and Mennucci [65, 66, 67]; more recently, Yezzi-M.-Sundaramoorthi[53–58, 35] have studied Sobolev-like metrics of curves and shown many goodproperties for applications to shape optimization; similar results have also beenshown independently by Charpiat et al. [9, 10, 11].We now discuss some Riemannian metrics on immersed curves.• The H 0 metric∫〈h 1 , h 2 〉 H 0 = 〈h 1 (s), h 2 (s)〉 ds .• Michor and Mumford [37]’s metric∫〈h 1 , h 2 〉 H A = (1 + A|κ c | 2 )〈h 1 , h 2 〉 ds .• Yezzi and Mennucci [67] conformal metric• Charpiat et al. [10] rigidified normscc∫〈h 1 , h 2 〉 H 0 = φ(c) 〈hφ1 , h 2 〉 ds .c58