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

• homotopy-wise parameterization invariant if, for any ϕ : [0, 1] ×S 1 → S 1 smooth, ϕ(t, ·) a diffeomorphism of S 1 for all fixed t, let ˜C(t, θ) =C(t, ϕ(t, θ)), then E( ˜C) = E(C).It is not difficult to prove that the second condition implies the first. Thereis an important remark to note: a homotopy-wise-parameterization-invariantRiemannian metric cannot be a proper metric.Proposition 11.8 Suppose that a Riemannian metric is homotopy-wise parameterizationinvariant; if h 1 , h 2 ∈ T c M and h 1 is tangent to c at all points, then〈h 1 , h 2 〉 c = 0. So the Riemannian metric in this case is actually a semimetric(and ‖ · ‖ c is not a norm, but rather a seminorm in T c M).Proof. Let C(t, u) = c(ϕ(t, u)) with ϕ(t, ·) be a time-varying family of diffeomorphismsof S 1 . So E(C) = E(c) = 0, that is∫ 10‖c ′ ∂ t φ‖ 2 c dt = 0so ‖c ′ ∂ t φ‖ c = 0; but note that, by choosing appropriately φ, we may representany h ∈ T c M that is tangent to c as h = c ′ ∂ t φ. By polarization, we obtain thethesis.11.6 Standard and geometric distanceThe standard distance d(c 0 , c 1 ) in M is the infimum of Len(γ) where γ is anyhomotopy that connects c 0 , c 1 ∈ M (cf. Definition 3.25).We are though interested in studying metrics and distances in the quotientspace B := M/Diff(S 1 ).We suppose that the metric G is “curve-wise parameterization invariant”;then G may be projected to B := M/Diff(S 1 ). So in the following we will use adifferent definition of “geometric distance”.Definition 11.9 (geometric distance) d G (c 0 , c 1 ) is the infimum of the lengthLen(C) in the class of all homotopies C connecting the curve c 0 to any reparameterizationc 1 ◦ φ of c 1 .This implements the quotienting formula that we saw in Definition 3.6 (inthis case, the group is G = Diff(S 1 )): so we will use d G as a distance on B.At the same time, note that in writing d G (c 0 , c 1 ) we are abusing notation:d G is not a distance in the space M, but rather it is a semidistance, since thedistance between c and a reparameterization c ◦ φ is zero.11.7 Horizontal and vertical spaceConsider a metric G (curve-wise parameterization invariant) on M. Let Πonce again be projection from M := Imm f (S 1 ) to the quotient B = B i,f =M/Diff(S 1 ).We present a list of definitions (see also the figure 19 on the next page).93