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

• sometimes S 1 will be identified with lR/(2π) (that is lR modulus 2π translations).• but other times (and in particular if the curve is planar) we will associateS 1 = {e it , t ∈ lR} = {z, |z| = 1} ⊂ IC.We recall that a curve is a map c : S 1 → lR n . The image of the curve isc(S 1 ).Definition 4.1 (Classes of Curves)• Imm(S 1 , lR n ) is the class of immersed curves c, such that c ′ ≠ 0at all points.• Imm f (S 1 , lR n ) is the class of freely immersed curve, the immersedcurves c such that, moreover, if φ : S 1 → S 1 is a diffeomorphism andc(φ(t)) = c(t) for all t, then φ =Id. So, in a sense, the curve is “completely”characterized by its image.• Emb(S 1 , lR n ) are the embedded curves, maps c that are diffeomorphiconto their image c(S 1 ); and the image is an embeddedsubmanifold of lR n of dimension 1.Each class contains the one following it.non-freely immersed curve.The following is an example of aExample 4.2 We define the doubly traversed circle using the complex notationc(z) = z 2 for z ∈ S 1 ⊂ IC; or otherwise identifying S 1 = lR/(2π), and inthis casec(θ) = (cos(2θ), sin(2θ))for θ ∈ lR/(2π). Setting φ(t) = t + π, we have that c = c ◦ φ, so c is not freelyimmersed.Vice versa, the following result is a sufficient condition to assert that a curve isfreely immersed.Proposition 4.3 (Michor and Mumford [37]) If c is immersed and thereis a x ∈ lR n s.t. c(t) = x for one and only one t, then c is freely immersed.Remark 4.4 Imm(S 1 , lR n ) , Imm f (S 1 , lR n ) , Emb(S 1 , lR n ) , are open subsetof the Banach space C r (S 1 → lR n ) when r ≥ 1, so they are trivially submanifolds(the charts are identity maps); and similarly for the Fréchet space C ∞ (S 1 → lR n ).Imm(S 1 , lR n ) , Imm f (S 1 , lR n ) , Emb(S 1 , lR n ) are connected iff n ≥ 3;whereas in the case n = 2 of planar curves, they are divided in connectedcomponents each containing curves with the same winding number (see Prop. 8.1for the definition).34