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

11.13 MomentaWhat follows comes from §2.5 in Michor and Mumford [36].Consider again the action of a group G on a Riemannian manifold M (asdefined in Sect. 3.8). Most groups that act on curves are smooth differentiablemanifolds themselves, and the group operations are smooth: so they are Liegroups. In our cases of interest, the actions are smooth as well.We want to study some differentiable properties of the action g · m; using therule (3.5), we see that we can restrict to studying the case e · m, where e ∈ G isthe identity element.11.13.1 Conservation of momentaGiven a curve c ∈ M, and a tangent vector ξ ∈ T e G (where T e G is the Liealgebra of G), we derive the action, for fixed c and e “moving” in direction ξ;the result of this derivative is a ζ = ζ ξ,c ∈ T c M (depending linearly on ξ). Thisdirection ζ is intuitively “the infinitesimal motion of c, when we infinitesimallyact in direction ξ on c”.To exemplify the above process, we provide a simple toy example.Example 11.22 (Rotation action on the plane) The group of 2D rotationsG = lR/(2π) (the real line modulus 2π, with the + operation) acts on the plane,as followsG × lR 2 → lR 2g, m ↦→ g · m = R g mwith( )cos g − sin gR g :=.sin g cos gSince T e G = lR, there is only one direction in the tangent to G, hence thederivative in direction ξ = 1 is just the standard derivative d/dg; similarlyT lR 2 = lR 2 ; by deriving the action in point g = e = 0 (the identity element), weobtain thatζ 1,m = JmwithJ :=( 0 −11 0Note that Jm is the vector (normal to m) obtained by rotating m of an angle π/2counterclockwise. The physical interpretation is that, when rotating the plane incounterclockwise direction with speed ξ = 1, each point m will move with velocityζ = Jm.In the above toy example, the result ζ is vector; but in the case of immersedcurves M, since ζ ∈ T c M, then ζ = ζ(θ) : S 1 → lR n is a vector field.).We then recall a very interesting condition by Emmy Noether99