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

Theorem 11.23 Suppose that the Riemannian metric 〈·, ·〉 c on M is invariantw.r.to the action of G: this is equivalent to saying that the action is an isometry.Let γ(t) be a critical geodesic: then〈 〉ζ ξ,γ(t) , ˙γ(t)(11.5)γ(t)is constant in t (for any choice of ξ ∈ T e G).The quantity (11.5) is called “the momentum of the action”.As an alternative interpretation, note that the vectors ζ that are obtainedby deriving the action are exactly all the vectors in the vertical spaces V c ofthe corresponding action G (since ζ are infinitesimal motions inside the orbit).Recall that h ∈ T c M is horizontal (that is, h ∈ W c ) iff it is orthogonal to V c〈ζ, h ′ 〉 = 0 for all ζ ∈ V c . So, as corollary of Emmy Noether’s theorem, we obtainthatCorollary 11.24 if a geodesic is shot in a horizontal direction ˙γ(0), then thegeodesic will be horizontal for all subsequent times.The following are examples of momenta that are related to the actions oncurves (that we saw in Example 4.8).Example 11.25 • The rescaling group is represented by lR + , that is onedimensional, so there is only one tangent direction ξ = 1 in lR + . Theaction is l, c ↦→ lc, so there is only one direction, that is ζ = c.• The translation group is represented by lR n , that is a vector space, so thetangent directions are ξ ∈ lR n ; the action is ξ, c ↦→ ξ + c; then ζ = ξ (andnote that this is constant in θ).• The rotation group is represented by orthogonal matrixes, so we set G =O(n); the group identity e is the identity matrix; the action is matrixvectormultiplication A, c(θ) ↦→ Ac(θ); the tangent T e G is the set of theantisymmetric matrixes B ∈ lR n×n ; then ζ = Bc.• The reparameterization group is G = Diff(S 1 ); a tangent vector is a scalarfield ξ : S 1 → lR; the action is the composition φ, c ↦→ c ◦ φ; we in the endhave thatζ(θ) = ξ(θ)c ′ (θ)(where c ′ = D θ c) that is, ζ is a generic vector field parallel to the curve.For the ˜H 1 metric, this implies the following properties of a geodesic path γ(t).Proposition 11.26 • Scaling momentum:∫〈γ, ˙γ〉 ˜H1= avg c (γ) · avg c ( ˙γ) + λL 2D s γ · D s ˙γ ds = constant.100