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

The term preshape space is sometimes used for the leftmost space, when bothspaces are studied in the same paper.4.5 Geometric curvesUnfortunately the quotientB i = Imm(S 1 , lR n )/Diff(S 1 )of immersed curves up to reparameterization is not a Fréchet manifold.We (re)define the space of geometric curves.Definition 4.9B i,f (S 1 , lR n ) = Imm f (S 1 , lR n )/Diff(S 1 )is the quotient of Imm f (S 1 , lR n ) (the free immersions) by the diffeomorphismsDiff(S 1 ) (that act as reparameterizations).The good news is thatProposition 4.10 (§2.4.3 in Michor and Mumford [37]) If Imm f has thetopology of the Fréchet space of C ∞ functions, then B i,f is a Fréchet manifoldmodeled on C ∞ .The bad news is that• when we add a simple Riemannian metric to B i,f , the resulting metricspace is not metrically complete; indeed, there cannot be any norm on C ∞that generates the same topology of the Fréchet space C ∞ (as we discussedin 3.26);• by modeling B i,f as a Fréchet manifold, some calculus is lost, as we saw inSection 3.2.5.Remark 4.11 It seems that this is the only way to properly define the manifold.If otherwise we choose M = C k (S 1 → lR n ) to be the manifold of curves, then ifc ∈ M, c ′ ∉ T c M. Hence we (must?) model M on C ∞ functions.4.5.1 Research pathFollowing Michor and Mumford [37] we so obtained a possible program of mathresearch:• defineB = B i,f (S 1 , lR n ) = Imm f (S 1 , lR n )/Diff(S 1 )and consider B as a Fréchet manifold modeled on C ∞ ,• define a Riemann/Finsler geometry on it, study its properties,• metrically complete the space.In the last step, we would hope to obtain a differentiable manifold; unfortunately,this is sometimes not true, as we will see in the overview of the literature.38