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

Theorem 3.30 Suppose that M is a smooth differentiable manifold modeledon a Hilbert space with a smooth Riemannian metric. The derivative of theexponential map exp c : T c M → M at the origin is an isometry, hence exp c is alocal diffeomorphism between a neighborhood of 0 ∈ T c M and a neighborhood ofc ∈ M.(See [30], VIII §5). The exponential map can then “linearize” small portions ofM, and so it will enable us to use linear methods such as the principal componentanalysis. Unfortunately, the above result does not hold if M is modeled on aFréchet space.3.6.6 Hopf–Rinow theoremTheorem 3.31 (Hopf–Rinow) Suppose M is a finite dimensional Riemannianor Finsler manifold. The following statements are equivalent:• (M, d) is metrically complete;• the O.D.E. (3.3) can be solved for all c ∈ M, η ∈ T c M and v ∈ lR;• the map exp cis surjective;and all those imply that, ∀x, y ∈ M there exists a minimal geodesic connecting xto y.3.6.7 Drawbacks in infinite dimensionsIn a certain sense, infinite dimensional manifolds are simpler than their correspondingfinite-dimensional counterparts: indeed, by Eells and Elworthy [18],Theorem 3.32 (Eells–Elworthy) Any smooth differentiable manifold M modeledon an infinite dimensional separable Hilbert space H may be embedded asan open subset of that Hilbert space.In other words, it is always possible to express M using one single chart. (Butnote that this may not be the best way for computations/applications).When M is an infinite dimensional Riemannian manifold, though, only asmall part of the Hopf–Rinow theorem still holds.Proposition 3.33 Suppose M is infinite dimensional, modeled on a Hilbertspace, and (M, d) is complete, then the O.D.E. (3.3) of a critical geodesic canbe solved for all v ∈ lR.But other implications fails.Example 3.34 (Atkin [3]) There exists an infinite dimensional complete Hilbertsmooth manifold M and x, y ∈ M such that there is no critical geodesic connectingx to y.30