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

3.6.1 Riemann metric, lengthDefinition 3.23 • A Riemannian metric G on a differentiable manifoldM defines a scalar product 〈h 1 , h 2 〉 G|c on h 1 , h 2 ∈ T c M, dependent onthe point c ∈ M. We assume that the scalar product varies smoothly w.r.toc.• The scalar product defines the norm |h| c = |h| G|c = √ 〈h, h〉 G|c.Suppose γ : [0, 1] → M is a path connecting c 0 to c 1 .∫ 1• The length is Len(γ) := | ˙γ(v)| γ(v) dv0γwhere ˙γ(v) := ∂ v γ(v).c 1∫ M1c 0• The energy (or action) is E(γ) := | ˙γ(v)| 2 γ(v) dv3.6.2 Finsler metric, lengthDefinition 3.24 We define a Finsler metric to be a function F : T M → lR + ,such that• F is continuous and,• for all c ∈ M , v ↦→ F (c, v) is a norm on T c M.We will sometimes write |v| c := F (c, v).(Sometimes F is called a “Minkowsky norm”).As for the case of norms, a Finsler metric may be asymmetric; but, for sakeof simplicity, we will only consider the symmetric case.Using the norm |v| c we can then again define the length of paths byLen F (γ) =and similarly the action.3.6.3 Distance∫ 10| ˙γ(t)| γ(t) dt =0∫ 10F (γ(t), ˙γ(t)) dtDefinition 3.25 The distance d(c 0 , c 1 ) is the infimum of Len(γ) between allC 1 paths γ connecting c 0 , c 1 ∈ M.Remark 3.26 In the following chapter, we will define some differentiable manifoldsM of curves, and add a Riemann (or Finsler) metric G on those; thereare two different choices for the model space,• suppose we model the differentiable manifold M on a Hilbert space U, withscalar product 〈, 〉 U ; this implies that M has a topology τ associated toit, and this topology, through the charts φ, is the same as that of U. Let28