Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ... 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
That is,• (M, d) is complete ⇏ exp c is surjective,• and (M, d) is complete ⇏ minimal geodesics exist.It is then, in general, quite difficult to prove that an infinite dimensionalmanifold admits minimal geodesics (even when it is known to be metricallycomplete). There are though some positive results, such as Ekeland [19] (thatwe cannot properly discuss for lack of space); or the following.Theorem 3.35 (Cartan–Hadamard) Suppose that M is connected, simplyconnected and has seminegative sectional curvature; then these are equivalent:• (M, d) is complete• there exists a c ∈ M such that the map η → exp c (η) is well definedand then there exists an unique minimal geodesic connecting any two points.For the proof, see Corollary 3.9 and 3.11 in sec. IX.§3 in Lang [30].3.7 Gradient in abstract differentiable manifoldLet M be a differentiable manifold, c ∈ M, and φ : U → V be a chart withc ∈ V .Definition 3.36 Let E : M → lR be a functional; we say that it is Gâteauxdifferentiable at c if for all h ∈ T c M there exists the directional derivativeat c in direction hwhere x = φ −1 (c), k = [Dφ(c)] −1 (h).DE(c; h) := limt→0E(φ(x + tk)) − E(c)t. (3.4)Fixing c (and then x), DE(c; ·) is a linear function from h ∈ T c M into lR;for this reason it is considered an element of the cotangent space T ∗ c M (thatis the dual space of T c M — and we will not discuss here further). Suppose nowthat we add a Riemannian geometry to M; this defines an inner product 〈·, ·〉 con T c M, so we can then define the gradient.Definition 3.37 (Gradient) The gradient ∇E(c) of E at c is the uniquevector v ∈ T c M such that〈v, h〉 c = DE(c; h) ∀h ∈ T c M .If M is modeled on a Hilbert space H, and the inner product 〈·, ·〉 c used onT c M is equivalent to the inner product in H (as we discussed in 3.26), then theabove equation uniquely defines what the gradient is. When M is modeled on aFréchet space, though, there is no choice of inner product that is “compatible”with M; and there are pathological situations where the gradient does not exist.31
- Page 1 and 2: Metrics of curves in shape optimiza
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- Page 7 and 8: κ > 0HNNHNHκ < 0Figure 1: Example
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- Page 15 and 16: 2.4.1 Example: geometric heat flowW
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That is,• (M, d) is complete ⇏ exp c is surjective,• <strong>and</strong> (M, d) is complete ⇏ m<strong>in</strong>imal geodesics exist.It is then, <strong>in</strong> general, quite difficult to prove that an <strong>in</strong>f<strong>in</strong>ite dimensionalmanifold admits m<strong>in</strong>imal geodesics (even when it is known to be metricallycomplete). There are though some positive results, such as Ekel<strong>and</strong> [19] (thatwe cannot properly discuss for lack <strong>of</strong> space); or the follow<strong>in</strong>g.Theorem 3.35 (Cartan–Hadamard) Suppose that M is connected, simplyconnected <strong>and</strong> has sem<strong>in</strong>egative sectional curvature; then these are equivalent:• (M, d) is complete• there exists a c ∈ M such that the map η → exp c (η) is well def<strong>in</strong>ed<strong>and</strong> then there exists an unique m<strong>in</strong>imal geodesic connect<strong>in</strong>g any two po<strong>in</strong>ts.For the pro<strong>of</strong>, see Corollary 3.9 <strong>and</strong> 3.11 <strong>in</strong> sec. IX.§3 <strong>in</strong> Lang [30].3.7 Gradient <strong>in</strong> abstract differentiable manifoldLet M be a differentiable manifold, c ∈ M, <strong>and</strong> φ : U → V be a chart withc ∈ V .Def<strong>in</strong>ition 3.36 Let E : M → lR be a functional; we say that it is Gâteauxdifferentiable at c if for all h ∈ T c M there exists the directional derivativeat c <strong>in</strong> direction hwhere x = φ −1 (c), k = [Dφ(c)] −1 (h).DE(c; h) := limt→0E(φ(x + tk)) − E(c)t. (3.4)Fix<strong>in</strong>g c (<strong>and</strong> then x), DE(c; ·) is a l<strong>in</strong>ear function from h ∈ T c M <strong>in</strong>to lR;for this reason it is considered an element <strong>of</strong> the cotangent space T ∗ c M (thatis the dual space <strong>of</strong> T c M — <strong>and</strong> we will not discuss here further). Suppose nowthat we add a Riemannian geometry to M; this def<strong>in</strong>es an <strong>in</strong>ner product 〈·, ·〉 con T c M, so we can then def<strong>in</strong>e the gradient.Def<strong>in</strong>ition 3.37 (Gradient) The gradient ∇E(c) <strong>of</strong> E at c is the uniquevector v ∈ T c M such that〈v, h〉 c = DE(c; h) ∀h ∈ T c M .If M is modeled on a Hilbert space H, <strong>and</strong> the <strong>in</strong>ner product 〈·, ·〉 c used onT c M is equivalent to the <strong>in</strong>ner product <strong>in</strong> H (as we discussed <strong>in</strong> 3.26), then theabove equation uniquely def<strong>in</strong>es what the gradient is. When M is modeled on aFréchet space, though, there is no choice <strong>of</strong> <strong>in</strong>ner product that is “compatible”with M; <strong>and</strong> there are pathological situations where the gradient does not exist.31
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