Riemannian metrics on Shape spaces of curves and surfaces

07/11/2017
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Riemannian metrics on Shape spaces of curves and surfaces Alice Barbara Tumpach Laboratoire Painlevé, Lille University, France GSI 2017 Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Outline Part I 1 What are the Model spaces of infinite-dimensional geometry? 2 What are the Tools from Functional Analysis? 3 What are the Toys we can play with? 4 What are the Traps of infinite-dimensional geometry? Part II A trap of infinite-dimensional geometry : there exist Poisson brackets that are not given by bivector fields Part III 1 Shape spaces as Quotient versus Sections of fiber bundles 2 Riemannian metrics on Shape spaces 3 Gauge invariant metrics on Shape spaces Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Model spaces of infinite-dimensional geometry? Hilbert ⊂ Banach ⊂ Fréchet ⊂ Locally Convex spaces Hilbert space H = complete vector space for the distance given by an inner product = h·, ·i : H × H → R+ symmetric :hx, yi = hy, xi bilinear : hx, y + λzi = hx, yi + λhx, zi non-negative : hx, xi ≥ 0 definite : hx, xi = 0 ⇒ x = 0 H∗ = H (Riesz Theorem). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Model spaces of infinite-dimensional geometry? Hilbert ⊂ Banach ⊂ Fréchet ⊂ Locally Convex spaces Banach space B = complete vector space for the distance given by a norm = k · k : B → R+ triangle inequality : kx + yk ≤ kxk + kyk absolute homogeneity : kλxk = |λ|kxk. non-negative : kxk ≥ 0 definite : kxk = 0 ⇒ x = 0. B∗ = Banach space. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Model spaces of infinite-dimensional geometry? Hilbert ⊂ Banach ⊂ Fréchet ⊂ Locally Convex spaces Fréchet space F = complete Hausdorff vector space for the distance d : F × F → R+ given by a countable family of semi-norms k · kn : d(x, y) = +∞ X n=0 1 2n kx − ykn 1 + kx − ykn F∗ 6= Fréchet space in general, but locally convex F∗∗ = Fréchet space. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Model spaces of infinite-dimensional geometry? Hilbert ⊂ Banach ⊂ Fréchet ⊂ Locally Convex spaces Locally Convex spaces = Hausdorff topological vector space whose topology is given by a (possibly not countable) family of semi-norms. References : Klingenberg : Riemannian Geometry Lang : Differential and Riemannian manifolds Fondamentals of Differential Geometry Hamilton :The inverse function theorem of Nash-Moser A. Kriegl and P. Michor : Convenient setting of Global Analysis Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Tools from Functional Analysis? Banach-Picard fixed point Theorem (E, d) complete metric space f : E → E contraction of E : d(f (x), f (y)) ≤ kd(x, y) where k ∈ (0, 1) ⇒  ∃ ! x ∈ E, f (x) = x ∀x0 ∈ E, the sequence xn+1 = f (xn) converges to x Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Tools from Functional Analysis? Hahn-Banach Theorem E locally convex space A ⊂ E a convex x ∈ E, x / ∈ A ⇒ ∃ continuous functional ` : E → R with `(x) / ∈ `(A) Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Tools from Functional Analysis? Open mapping Theorem  F Fréchet G Fréchet or  F webbed locally convex G inductive limit of Baire locally convex spaces L : F → G continuous, linear, and surjective ⇒ L is open Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Tools from Functional Analysis? Inverse function Theorem Theorem Let f : U ⊂ B1 → B2 be a C 1 -map between Banach spaces. If Df (a) is invertible at a ∈ U , then there exists an open neighborhood Va of a ∈ U and an open neighborhood Vf (a) ⊂ B2 suth that f : Va → Vf (a) is a C 1 -diffeomorphism. Counterexample : exp : Lie(Diff(S1 )) → Diff(S1 ) not locally onto. Theorem (Nash-Moser) Let f : U ⊂ F1 → F2 be a smooth tame map between Fréchet spaces. Suppose that the equation for the derivative Df (x)(h) = k has a unique solution h = L(x)k for all x ∈ U and ∀k ∈ F2 and that the family of inverses L : U × F2 → F1 is a smooth tame map. Then f is locally invertible and each local inverse is a smooth tame map. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Tools from Functional Analysis? Theorems : Hilbert Banach Fréchet Locally Convex Banach-Picard √ √ √ X Open Mapping √ √ √ F webbed G limit of Baire Hahn-Banach √ √ √ √ Inverse function √ √ Nash-Moser X Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Toys we can play with? Riemannian ⊂ Symplectic ⊂ Poisson Geometry Riemannian metric = smoothly varying inner product on a manifold M gx : Tx M × Tx M → R (U, V ) 7→ gx (U, V ) strong Riemannian metric = for every x ∈ M, gx : Tx M → (Tx M)∗ is an isomorphism weak Riemannian metric = for every x ∈ M, gx : Tx M → (Tx M)∗ is just injective Levi-Cevita connection may not exist for a weak Riemannian metric. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Toys we can play with? Riemannian ⊂ Symplectic ⊂ Poisson Geometry Symplectic form = smoothly varying skew-symmetric bilinear form ωx : Tx M × Tx M → R (U, V ) 7→ ωx (U, V ) with dw = 0 and (Tx M)⊥w = {0} strong symplectic form = for every x ∈ M, ωx : Tx M → (Tx M)∗ is an isomorphism weak symplectic form = for every x ∈ M, ωx : Tx M → (Tx M)∗ is just injective Darboux Theorem does not hold for a weak symplectic form Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Toys we can play with? Riemannian ⊂ Symplectic ⊂ Poisson Geometry Hamiltonian Mechanics (M, g) strong Riemannian manifold [ : Tx M ' T∗ x M [−1 = ] U 7→ gx (U, ·) Kinetic energy = Hamiltonian H : T∗ M → R ηx 7→ gx (η] x , η] x ) (T∗ M, ω) strong symplectic manifold π : T∗ M → M ω = dθ θ(x,η) : Tx,ηT∗ M → R Liouville 1-form X 7→ η(π∗(X)) geodesic flow = flow of Hamiltonian vector field XH : dH = ω(XH , ·) Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Toys we can play with? Riemannian ⊂ Symplectic ⊂ Poisson Geometry Poisson bracket = family of bilinear maps {·, ·}U : C ∞ (U) × C ∞ (U) → C ∞ (U), U open in M with skew-symmetry {f , g}U = −{g, f }U Jacobi identity {f , {g, h}U }U + {g, {h, f }U }U + {h, {f , g}U }U = 0 Leibniz rule {f , gh}U = {f , g}U h + g{f , h}U A strong symplectic form defines a Poisson bracket by {f , g} = ω(Xf , Xg ) where df = ω(Xf , ·) and dg = ω(Xg , ·) A Poisson bracket may not be given by a bivector field Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the Toys we can play with? Riemannian Symplectic Complex    ⊂ Kähler ⊂ hyperkähler Geometry Complex structure = smoothly varying endomorphism J of the tangent space s.t. J2 = −1. Integrable complex structure : s. t. there exists an holomorphic atlas Formally integrable complex structure : with Nijenhuis tensor = 0 Newlander-Nirenberg Theorem is not true in general : formal integrability does not imply integrability. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps What are the traps of infinite-dimensional geometry? In infinite-dimensional geometry, the golden rule is : "Never believe anything you have not proved yourself!" The distance function associated to a Riemannian metric may by the zero function (Example by Michor and Mumford). Levi-Cevita connection may not exist for weak Riemannian metrics Hopf-Rinow Theorem does not hold in general : geodesic completeness 6= metric completeness Darboux Theorem does not apply to weak symplectic forms A formally integrable complex structure does not imply the existence of a holomorphic atlas the tangent space differs from the space of derivations (even on a Hilbert space) a Poisson bracket may not be given by a bivector field (even on a Hilbert space) Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Model spaces Tools Toys Traps arXiv:1710.03057v1 [math.FA] 9 Oct 2017 QUEER POISSON BRACKETS DANIEL BELTITA Institute of Mathematics “S. Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania TOMASZ GOLIŃSKI University in Białystok, Institute of Mathematics, Ciołkowskiego 1M, 15-245 Białystok, Poland ALICE-BARBARA TUMPACH Université de Lille, Laboratoire Painlevé, CNRS U.M.R. 8524, 59 655 Villeneuve d’Ascq Cedex, France Abstract. We give a method to construct Poisson brackets { · , · } on Banach manifolds M, for which the value of {f, g} at some point m ∈ M may depend on higher order derivatives of the smooth functions f, g: M → R, and not only on the first-order derivatives, as it is the case on all finite-dimensional manifolds. We discuss specific examples in this connection, as well as the impact on the earlier research on Poisson geometry of Banach manifolds. 1. Introduction The Poisson brackets in infinite-dimensional setting have played for a long time a significant role in various areas of mathematics including classical me- chanics and integrable systems theory (see e.g. [Fad80]). However the rigor- ous approach to the notion of Poisson manifold in the context of Banach space is relatively new (see e.g. [AMR02] and [OR03]). It is known that the Poisson brackets on infinite-dimensional manifolds lack some of the properties known from the finite-dimensional case. It was shown for instance in [OR03] that the E-mail addresses: beltita@gmail.com, tomaszg@math.uwb.edu.pl, Barbara.Tumpach@math.univ-lille1.fr. 1 Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Poisson bracket not given by a Poisson tensor Poisson bracket not given by a Poisson tensor H separable Hilbert space Kinetic tangent vector X ∈ Tx H equivalence classes of curves c(t), c(0) = x, where c1 ∼ c2 if they have the same derivative at 0 in a chart. Operational tangent vector x ∈ H is a linear map D : C∞ x (H ) → R satisfying Leibniz rule : D(fg)(x) = Df g(x) + f (x) Dg Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Poisson bracket not given by a Poisson tensor Poisson bracket not given by a Poisson tensor Ingredients : Riesz Theorem Hahn-Banach Theorem compact operators K (H ) ( B(H ) bounded operators ⇒ ∃` ∈ B(H )∗ such that `(id) = 1 and `| K (H ) = 0. Queer tangent vector [Kriegl-Michor] Define Dx : C∞ x (H ) → R, Dx (f ) = `(d2 (f )(x)), where the bilinear map d2 (f )(x) is identified with an operator A ∈ B(H ) by Riesz Theorem d2 (f )(x)(X, Y ) = hX, AY i Then Dx is an operational tangent vector at x ∈ H of order 2 Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Poisson bracket not given by a Poisson tensor Poisson bracket not given by a Poisson tensor Queer tangent vector [Kriegl-Michor] d(fg)(x)(X) = df (x)(X).g(x) + f (x).dg(x)(X) d2 (fg)(x)(X, Y ) = d2 f (x)(X, Y ).g(x) + df (x)(X)dg(x)(Y ) +df (x)(Y )dg(x)(X) + f (x)d2 g(x)(X, Y ) d2 (fg)(x) = d2 f (x).g(x) + `(df (x) ⊗ dg(x)) +`(dg(x) ⊗ df (x)) + f (x)d2 g(x)(X, Y ) Dx (fg) = `(d2 (fg)(x)) = `(d2 f (x)).g(x) + f (x)`(d2 g(x)) +`(df (x) ⊗ dg(x)) + `(dg(x) ⊗ df (x)) = Dx f g(x) + f (x) Dx g Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Poisson bracket not given by a Poisson tensor Poisson bracket not given by a Poisson tensor Theorem Consider M = H × R. Denote points of M as (x, λ). Then {·, ·} defined by {f , g}(x, λ) := Dx (f (·, λ)) ∂g ∂λ (x, λ) − ∂f ∂λ (x, λ)Dx (g(·, λ)) a queer Poisson bracket on H × R, in particular it can not be represented by a bivector field Π : T∗ M × T∗ M → R. The Hamiltonian vector field associated to h(x, λ) = −λ is the queer operational vector field Xh = {h, ·} = Dx acting on f ∈ C∞ x (H ) by Dx (f ) = `(d2 (f )(x)). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Shape spaces Pre-shape space F := {f immersion : S1 → R2 } ⊂ C ∞ (S1 , R2 ) Shape space S := 1-dimensional immersed submanifolds of R2 Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Shape spaces Pre-shape space F := {f embedding : S2 → R3 } ⊂ C ∞ (S2 , R3 ) Shape space S := 2-dimensional submanifolds of R3 Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Shape spaces are non-linear manifolds Figure: First line : linear interpolation between some parameterized ballerinas, second line : linear interpolation between arc-length parameterized ballerinas. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics For I = [0, 1] or I = Z/R ' S1 , the space of smooth immersions C(I) = ∞ \ k=1 Ck (I) = {γ ∈ C ∞ (I, R2 )/R2 , γ0 (s) 6= 0, ∀s ∈ I}. is an open set of C ∞ (I, R2 )/R2 for the topology induced by the family of norms k·kC k , hence a Fréchet manifold. C1(I) = {γ ∈ C(I) : Z 1 0 |γ0 (s)|ds = 1}. A1(I) = {γ ∈ C(I) : |γ0 (s)| = 1, ∀s ∈ I} ⊂ C1(I). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Theorem (A.B.T, S.Preston) The subset C1(I) is a tame C ∞ -submanifold of C(I) and A1(I) is a tame C ∞ -submanifold of C(I), and thus also of C1(I). Its tangent space at a curve γ is TγA1 = {w ∈ C ∞ (S1 , R2 ), w0 (s) · γ0 (s) = 0, ∀s ∈ S1 }. Proof : Uses the implicit function theorem of Nash-Moser. G (I) = Diff+ ([0, 1]) or Diff+ (S1 ) is a tame Fréchet Lie group [Hamilton]. Theorem (A.B.T, S.Preston) The right action Γ: C(I) × G (I) → C(I), Γ(γ, ψ) = γ ◦ ψ of the group of reparameterizations G (I) on the tame Fréchet manifold C(I) is smooth and tame, and preserves C1(I). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Theorem (A.B.T, S.Preston) Given a curve γ ∈ C1(I), let p(γ) ∈ A1(I) denote its arc-length-reparameterization, so that p(γ) = γ ◦ ψ where ψ0 (s) = 1 |γ0 ψ(s)  | , ψ(0) = 0. (1) Then p is a smooth retraction of C1(I) onto A1(I). Theorem (A.B.T, S.Preston) A1([0, 1]) is diffeomorphic to the quotient Fréchet manifold C1([0, 1])/G ([0, 1]). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Riemannian metrics We will consider the 2-parameter family of elastic metrics on C1(I) introduced by Mio et al. : Ga,b (w, w) = R 1 0  a (Dsw · v) 2 + b (Dsw, n) 2  |γ0 (t)| dt, (2) where a and b are positive constants, γ is any parameterized curve in C1(I), w is any element of the tangent space Tγ C1(I), with Dsw = w0 |γ0| denoting the arc-length derivative of w, v = γ0 /|γ0 | and n = v⊥ . Since the reparameterization group preserves the elastic metric Ga,b , it defines a quotient elastic metric on the quotient space C1([0, 1])/G ([0, 1]), which we will denote by G a,b . G a,b ([w], [w]) = inf u∈Tγ O Ga,b (w + u, w + u) Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: First line : linear interpolation between some parameterized ballerinas, second line : linear interpolation between arc-length parameterized ballerinas. Geodesic between some parameterized ballerinas with 200 points using Qmap : execution time = 350 s. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Since A1([0, 1]) is diffeomorphic to the quotient Fréchet manifold C1([0, 1])/G ([0, 1]), we can pull-back the quotient elastic metric G a,b to the space of arc-length parameterized curves A1([0, 1]) and define e Ga,b (w, w) = Ga,b ([w], [w]) = inf u∈Tγ O Ga,b (w + u, w + u) where w is tangent to A1([0, 1]). If Tγ C1([0, 1]) decomposes as Tγ C1([0, 1]) = TγO ⊕ Horγ, this minimum is achieved by the unique vector Ph(w) ∈ [w] belonging to the horizontal space Horγ at γ. In this case: e Ga,b (w, w) = Ga,b (Ph(w), Ph(w)), (3) where Ph(w) ∈ Tγ C1([0, 1]) is the projection of w onto the horizontal space. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Theorem Let w be a tangent vector to the manifold A1([0, 1]) at γ and write w0 = Φ n, where Φ is a real function in C ∞ ([0, 1], R). Then the projection Ph(w) of w ∈ TγA1([0, 1]) onto the horizontal space Horγ reads Ph(w) = w − m v where m ∈ C ∞ ([0, 1], R) is the unique solution of − a b m00 + κ2 m = κΦ, m(0) = 0, m(1) = 0 (4) where κ is the curvature function of γ. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Toy example: initial path joining a circle to the same circle via an ellipse. The 5 first shapes at the left correspond to the path at time t = 0, t = 0.25, t = 0.5, t = 0.75 and t = 1. The right picture shows the entire path, with color varying from red (t = 0) to blue (t = 0.5) to red again (t = 1). a/b = 1/4 a/b = 0.01 a/b = 5 a/b = 1 a/b = 100 a/b = 50 a/b = 30 a/b = 20 a/b = 13 a/b = 10 Figure: Gradient of the energy functional at the middle of the path depicted in Fig. 3 for b = 1 and different values of the parameter a/b. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Toy example: initial path joining a circle to the same circle via an ellipse. The 5 first shapes at the left correspond to the path at time t = 0, t = 0.25, t = 0.5, t = 0.75 and t = 1. Figure: Gradient of the energy functional at the middle of the path connecting a circle to the same circle via an ellipse for different values of the eccentricity of the middle ellipse. The first line corresponds to the values of parameters a = 0.01 and b = 1. The second line corresponds to a = 100 and b = 1. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Gradient of the energy functional along the path depicted in Fig. 3 for a = 1 (upper line), a = 5 (middle line) and a = 50 (lower line) and b = 1. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: 2-parameter family of variations of the middle shape of a path connecting a circle to the same circle Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics 0 10 20 30 40 50 60 70 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0 10 20 30 40 50 60 70 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Figure: Energy functional for the 2-parameter family of paths whose middle shape is one of the shapes depicted in Fig. 8. The left upper picture corresponds to a = 0.01, b = 1 and the right upper picture to a = 100, b = 1. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics 0 10 20 30 40 50 60 70 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Figure: Energy functional for the 2-parameter family of paths whose middle shape is one of the shapes depicted in Fig. 8. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Scalar product on the tangent plan to the tip of the middle finger of a hand. Genus-0 surfaces of R3 are Riemann surfaces. Since they are compact and simply connected, the Uniformization Theorem says that they are conformally equivalent to the unit sphere. This means that, given a spherical surface, there exists a homeomorphism, called the uniformization map, which preserves the angles and transforms the unit sphere into the surface. The uniformization maps are parameterized by PSL(2, C). Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics A.B.Tumpach, H. Drira, M. Daoudi, A. Srivastava, Gauge invariant Framework for shape analysis of surfaces, IEEE TPAMI. A.B.Tumpach, Gauge invariance of degenerate Riemannian metrics, Notices of AMS. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Pairs of paths projecting to the same path in Shape space, but with different parametrizations. The energies of these paths, as computed by our program, are respectively (from the upper row to the lower row): E∆ = 225.3565, E∆ = 225.3216, E∆ = 180.8444, E∆ = 176.8673. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics Figure: Four Paths connecting the same shape but with a parametrization depending smoothly on time. The energy computed by our program is respectively E∆ = 0 for the path of hands, E∆ = 0.1113 for the path of horses, E∆ = 0 for the path of cats, and E∆ = 0.0014 for the path of Centaurs. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces Quotient versus Fiber bundle Riemannian metrics Gauge Invariant metrics D. Beltita, T. Golinski, A.B.Tumpach, Queer Poisson Brackets, ArXiv. A.B.Tumpach, S. Preston, Quotient elastic metrics on the manifold of arc-length parameterized plane curves, Journal of Geometric Mechanics. A.B.Tumpach, Gauge invariance of degenerate Riemannian metrics, Notices of AMS. A.B.Tumpach, H. Drira, M. Daoudi, A. Srivastava, Gauge invariant Framework for shape analysis of surfaces, IEEE TPAMI. D. Beltita, T. Ratiu, A.B. Tumpach, The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, Journal of Functional Analysis. A.B.Tumpach, Hyperkähler structures and infinite-dimensional Grassmannians, Journal of Functional Analysis. A.B.Tumpach, Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits, Annales de l’Institut Fourier. A.B.Tumpach, Classification of infinite-dimensional Hermitian-symmetric affine coadjoint orbits, Forum Mathematicum. Alice Barbara Tumpach Riemannian metrics on Shape spaces of curves and surfaces