Résumé

Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Graßmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

Rolling Symmetric Spaces

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application/pdf Rolling Symmetric Spaces Krzysztof Krakowski, Luís Machado, Fátima Leite

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            <title>Rolling Symmetric Spaces</title></titles>
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Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Graßmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

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Rolling Symmetric Spaces Lu´ıs Machado Department of Mathematics Institute of Systems and Robotics INSTITUTO DE SISTEMAS E ROBÓTI Departamento de Engenharia Electrotécnica e de Computadores* U Pólo II * 3030 – 290 Coimbra * Portugal Telef. +351 – 239 796 200/1 Fax. +351 – 239 406 672 Nº. Cont. (VAT) 502 854 227 Univ. of Tr´as-os-Montes e Alto Douro Univ. of Coimbra (Joint work with Krzysztof A. Krakowski and F´atima Silva Leite) 2nd Conference on Geometric Science of Information Ecole Polytechnique, Paris-Saclay (France) October 28-30, 2015 L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Main goals Present a unifying theory to describe rolling motions of Riemannian symmetric homogeneous spaces Establish the connection between the kinematic equations of rolling and the Lie algebra decomposition associated to the symmetric space L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Outline 1 Definition of Rolling 2 Homogenous Symmetric Spaces 3 Rolling Riemannian Symmetric Homogeneous Spaces 4 Illustrative examples Euclidean sphere Lorentzian sphere Grassmann manifold 5 Conjecture L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling map M Riemannian m−manifold (M = Rm ) G group of isometries on M (G = SO(m) ⋉ Rm ) M and M0 n−manifolds isometrically embedded in M Definition [Sharpe (1997)] χ : I ⊂ R −→ G is a rolling map of M on M0 along a smooth curve σ : I −→ M (rolling curve) if, for all t ∈ I, rolling: χ(t) · σ(t) := σ0(t) ∈ M0 (development curve of σ) Tχ(t)·σ(t) χ(t) M = Tχ(t)·σ(t)M0. no-slip: ˙σ0(t) = dσ(t)χ(t) · ˙σ(t) no-twist: dσ0(t) ˙χ(t)χ(t)−1 Tσ0(t)M0 ⊂ T⊥ σ0(t)M0 dσ0(t) ˙χ(t)χ(t)−1 T⊥ σ0(t)M0 ⊂ Tσ0(t)M0. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Illustration of rolling M0 σ0 σ M χ χ(t) M L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Structure of d ˙χχ−1 For an appropriate orthonormal basis of Tσ0(t)M Tσ0(t)M = Tσ0(t)M0 ⊕ T⊥ σ0(t)M0 Tσ0(t)M0 T ⊥ σ0(t)M0 dσ0(t) ˙χ(t)χ(t)−1 = 0 Xn×(m−n) −XT (m−n)×n 0 Tσ0(t)M0 T ⊥ σ0(t)M0 xxx We will show that on a symmetric space M, the structure of d ˙χχ−1 is captured from the decomposition of the Lie algebra associated to M. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Homogenous and symmetric spaces (g, p) −→ g · p ∈ M transitive action of G on M H = g ∈ G : g · p0 = p0 isotropy group at p0 ∈ M G/H is a homogenous space isomorphic to M Reductive homogenous spaces and symmetric spaces The homogenous space G/H is reductive if there exists a subspace p of g such that g = h ⊕ p, where h is the Lie algebra of H and h, p ⊆ p. Moreover, if p, p ⊆ h, G/H is a homogenous symmetric space. πp0 : g −→ g · p0 ∈ M Riemannian submersion Tp0 M ≃ g/h ≃ p L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Properties of the Rolling maps χ = ? Properties of the Rolling maps Symmetry χ1 χ−1 1 Properties of the Rolling maps Transitivity χ−1 2 L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Properties of the Rolling maps Transitivity χ = χ−1 2 ◦ χ1 Rolling symmetric homogeneous spaces Assume that M is the Euclidean space Rm M Riemannian symmetric homogeneous space G/H embedded in Rm (H isotropy group at p0) G = SO(m) ⋉ Rm group of isometries of Rm t → χ(t)= R(t), s(t) ∈SO(m)⋉ Rm rolling map of M upon Taff p0 M Action of χ = (R, s) on M and on TpM χ(t) · p = R(t) · p + s(t) dpχ(t) · V = R(t) · V L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Main results M Riemannian symmetric homogeneous space G/H Proposition If g = h ⊕ p (h is the Lie algebra of H), then h Tp0 M ⊆ Tp0 M h T⊥ p0 M ⊆ T⊥ p0 M and p Tp0 M ⊆ T⊥ p0 M p T⊥ p0 M ⊆ Tp0 M Theorem If g = h ⊕ p and χ is the rolling map of M = G/H upon Taff p0 M, then d ˙χχ−1 is an element of p. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 The n−sphere Sn Sn ≃ SO(n + 1)/SO(n) is a homogeneous symmetric space under the natural action of SO(n + 1): (Θ, p) → Θp The isotropy group at p0 = (0, . . . , 0, −1): H = Θ 0 0 1 : Θ ∈ SO(n) ≃ SO(n) so(n + 1) = h ⊕ p h = Ω 0 0 0 : Ω ∈ so(n) p = 0 v −vT 0 : v ∈ Rn L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the n−sphere Sn It is well known that the kinematic equations for the rolling sphere are ˙s(t) = A(t)p0 ˙R(t) = −A(t)R(t) , where A(t) = 0 v(t) −v(t)T 0 . [Jurdjevic (1997)] L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the n−sphere Sn It is well known that the kinematic equations for the rolling sphere are ˙s(t) = A(t)p0 ˙R(t) = −A(t)R(t) , where A(t) = 0 v(t) −v(t)T 0 . Then, xx d ˙χχ−1 ) = ˙RR−1 = −A ∈ p. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 The Lorentzian sphere Sn,1 For J =diag(1, . . . , 1, −1), consider the pseudo-orthogonal Lie group SO(n, 1)= X ∈ R(n+1)×(n+1) : XT JX = J ∧ det(X) = 1 Sn,1 ≃ SO(n, 1)/SO(n − 1, 1) is a homogeneous symmetric space under the natural action of SO(n, 1): (X, p) → Xp The isotropy group at p0 = (1, 0, . . . , 0): H = 1 0 0 Θ : Θ ∈ SO(n − 1, 1) ≃ SO(n − 1, 1) so(n, 1) = h ⊕ p h = 0 0 0 Ω : Ω ∈ so(n − 1, 1) p = 0 −vT v 0 J : v ∈ Rn L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the Lorentzian sphere The kinematic equations for the rolling Lorentzian sphere are ˙s(t) = u(t) ˙R(t) = p0 u(t)T − u(t) pT 0 JR(t) , where u(t) ∈ Tp0 Sn,1 . [Jurdjevic & Zimmerman (2008)] and [Korolko & Silva Leite (2011)] L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the Lorentzian sphere The kinematic equations for the rolling Lorentzian sphere are ˙s(t) = u(t) ˙R(t) = p0 u(t)T − u(t) pT 0 JR(t) , where u(t) ∈ Tp0 Sn,1 . Then, xx d ˙χχ−1 ) = ˙RR−1 = p0 uT − u pT 0 )J ∈ p. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 The Grassmann manifold Gn,k = P ∈ s(n) : P2 = P ∧ rank(P) = k Gn,k ≃ SO(n)/SO(k) × SO(n − k) is a homogeneous symmetric space under the action of SO(n): (X, P) → XPXT The isotropy group at P0 = Ik 0 0 0 : H = X1 0 0 X2 : X1 ∈ SO(k), X2 ∈ SO(n − k) ≃ SO(k) × SO(n − k) so(n) = h ⊕ p h = Ω1 0 0 Ω2 : Ω1 ∈ so(k), Ω2 ∈ so(n − k) p = 0 V −V T 0 : V ∈ Rk×(n−k) L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the Grassmann manifold The kinematic equations for the rolling of the Grassmann manifold are ˙X(t) = 0 Ψ(t) Ψ(t)T 0 ˙R(t) = 0 Ψ(t) −Ψ(t)T 0 R(t) , where Ψ(t) ∈ Rk×(n−k) . [H¨uper & Silva Leite (2007)] L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Rolling the Grassmann manifold The kinematic equations for the rolling of the Grassmann manifold are ˙X(t) = 0 Ψ(t) Ψ(t)T 0 ˙R(t) = 0 Ψ(t) −Ψ(t)T 0 R(t) , where Ψ(t) ∈ Rk×(n−k) . Then, xx d ˙χχ−1 = ad ˙RR−1 ∈ p. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015 Conjecture xx If M = G/H is a Riemannian symmetric homogenous space and g = h ⊕ p (h is the Lie algebra of H), then p Tp0 M ⊆ T⊥ p0 M, p T⊥ p0 M ⊆ Tp0 M and therefore d ˙χχ−1 ∈ p, even when the ambient manifold M is non-Euclidean. L. Machado, K. Krakowski, F. Silva Leite Rolling Symmetric Spaces GSI’2015, October 28-30, 2015