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

Definition 11.10 • The orbit is O c = [c] = {c ◦ φ | φ} = Π −1 ({c}).• The vertical space V c is the tangent to O c :V c := T c O c ⊂ T c Mthat can be explicitly written asV c := {h = b(s)c ′ (s) | b : S 1 → lR}i.e. all the vector fields h where h(s) is tangent to c.• The horizontal space W c is the orthogonal complement inside T c MW c := V ⊥c .Note that W c depends on G, but V c does not.MO cWccVcFigure 19: The action of Diff(S 1 ) on M. The orbits O c are dotted, the spacesW c and V c are dashed.The figure 20 may also help in understanding. The whole orbit O c is projectedto [c]. The vertical space V c is the kernel of DΠ c , so it is projected to 0.11.8 From curve-wise parameterization invariant to homotopywiseparameterization invariantIn the following sections, we just aim to give an overview of the theory; we willnot dwelve in depicting the most general hypotheses such that all presented resultsdo hold for a generic metric G; we will though, case by case, present referencesto how the results can be proven for the metrics discussed in this paper, and inparticular the Sobolev-type metrics.We suppose that this theorem holds.Theorem 11.11 Given any h ∈ T c M, there is an unique minimum point forminx∈W c‖x − h‖ Gand the minimum is called the horizontal projection of h.94