Riemannian and Sub-Riemannian Geodesic Flows

Transcription

1 Riemannian and Sub-Riemannian Geodesic Flows Mauricio Godoy Molina 1 Joint with E. Grong Universidad de La Frontera (Temuco, Chile) Potsdam, February Partially funded by grant Anillo ACT 1415 PIA CONICYT (Chile) Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

2 Outline 1 Introduction: Geometry and restrictions 2 Geodesic Flows 3 Results Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

3 Philosophy: Restrictions in mechanics In classical analytic mechanics we study systems of the form Differential equation + restrictions. Usually: restrictions on the position and/or velocity, but restr. on the velocity can hide restr. on the position. Definition (heuristic) Restrictions on the position (hidden or not) are called holonomic. In what follows, we call non-holonomic to those restrictions on velocity not hiding restrictions on the position. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

4 Hidden restrictions In examples 1-3: a particle P moves in R 3 with position r = r (t), velocity v = v (t) and under certain restrictions. Example 1: v r P moves on the sphere of radius r (0) centered at 0. In other words, v r is holonomic since it hides a restriction on the position. Definition The restriction on v can be integrated to a restriction on r. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

5 Perturbing restrictions Example 2: v k v = a ı + b j P moves on a horizontal plane. Holonomic again. But... what if we modify the problem slightly? Example 3: v ( k x j ) v = a ı + b ( j + x k ). Non-holonomic! Moreover: Theorem (Folklore) Small perturbations of restrictions are non-holonomic. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

6 Mathematical formulation Previous fact: Most restrictions are non-holonomic. That s why we want to understand them. Space of configurations: Q, with dim Q = dof. Restrictions: v (t) span{x 1,..., X k }. Example 1: Q = sphere, v (t) span { θ, ϕ } in spherical coordinates. Example 2: Q = horizontal plane, v (t) span { ı, j }. Example 3: Q = R 3, v (t) span { ı, j + x k }. Definition The flow of X Γ(Q) is the function Q R Q given by e τx p = solution of { γ(s) = X(γ(s)) γ(0) = p in time s = τ. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

7 Detecting holonomy I Flow of ı : e τ ı (x, y, z) = (x + τ, y, z). Flow of j : e τ j (x, y, z) = (x, y + τ, z). Flow of j + x k : e τ( j +x k ) (x, y, z) = (x, y + τ, z + τx). Theorem (Frobenius, 1877) A restriction H = span{x 1,..., X k } is holonomic iff d dτ e τy e τx e τy e τx p H, X, Y H, p Q. τ=0 Explanation? Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

8 Detecting holonomy II For simplicity: t = τ. Suppose X and Y restrictions in H. We follow X, afterward Y, Is the velocity of this new curve a restriction? If yes, then H is holonomic. we come back using X, finally we come back using Y. Definition The process described above is called commuting flows. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

9 Detecting holonomy III Example 2: Holonomic d dτ e τ j e τ ı e τ j e τ ı (x, y, z) = τ=0 d dτ (x, y, z) = (0, 0, 0) span { ı, } j. τ=0 }{{} constant! Example 3: Non-holonomic d dτ e τ( j +x k ) e τ ı e τ( j +x k ) e τ ı (x, y, z) = τ=0 d dτ (x, y, z + τ) = (0, 0, 1) = k / span { ı, } j + x k. τ=0 Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

10 Outline 1 Introduction: Geometry and restrictions 2 Geodesic Flows 3 Results Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

11 Affine control systems Given H = span{x 1,..., X k }, k n, we have the control problem k ẋ = u i X i (x) ẋ H. Same question: i=1 Given x 0, x T M, can we find ū 1,..., ū k such that x(0) = x 0 y x(t ) = x T? (SAC) BIG observation: The controllability of (SAC) is a consequence of H being completely non-holonomic. Definition H is completely non-holonomic if we can obtain any velocity by commuting flows in H. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

12 Geodesics Suppose that (SAC) is controllable. k ẋ = u i X i (x) ẋ H Problem i=1 Given x 0, x T M, we want to find ū = (ū 1,..., ū k ) such that x(0) = x 0, x(t ) = x T and 1 ( J(ū) = min J(u) = min u 1 (t) u k (t) 2) dt. u u 2 Usually: A curve γ with control ū is called sr-geodesic if k < n and R-geodesic if k = n. (SAC) Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

13 Cometric and metric Consider a bilinear non-negative tensor h on T M (a cometric) and h : T M T M = TM, p h (p, ) H = im h are the restrictions on M endowed with the metric h( h p, h q) = h (p, q) Fact (H, h) determines uniquely h. Besides ker h = Ann H. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

14 Hamiltonian and flows The Hamiltonian H h associated to the metric h is simply H h (p) = 1 2 h (p, p). If ω is the canonical 2-form on T M, then Definition The Hamiltonian vector field H h is given by dh h (X) = ω( H h, X), X X(M). If h is a (sub-)riemannian metric, then e t H h geodesic flow. is the (sub-)riemannian Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

15 Outline 1 Introduction: Geometry and restrictions 2 Geodesic Flows 3 Results Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

16 Motivating problem I Theorem (Montgomery) Given a principal G-bundle G M π N (+ technical hypotheses), then the normal sub-riemannian geodesics on M defined by H = (ker dπ) are given by γ sr (t) = exp r (tv) exp G ( ta(v)), where A is the g-valued connection one form. Idea: Compute Riemannian geodesics in M. Project down to N. Horizontally lift the curve to M. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

17 Horizontal lift Given a submersion π : M N and a vector v T x N, then at each x π 1 (x) there is a unique vector v T x M such that d x π( v) = v. This is the horizontal lift of v. For a vector field X Γ(TN), define X by X x = X x. The horizontal lift γ of γ : [0, T ] N at 2 x is the unique solution to γ = γ, γ(0) = x 2 Obviously π x = γ(0) Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

18 Motivating problem II Montgomery s theorem can be used for some examples. Hopf fibration: U(1) S 2n+1 CP 1 Quaternionic Hopf fibration: Sp(1) S 4n+3 HP n Grassmannians: U(n) V n,k Gr n,k BUT there are true fibrations: S 7 S 15 OP 1 = S 8 (octonionic Hopf). What to do when there is no group? Remark (Ornea, Parton, Piccinni, Vuletescu 2013) The situation is worse than expected: ANY v.f. tangent to the leaves of S 7 S 15 S 8 has a zero. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

19 Taming metrics Let (M, H, h) a sub-riemannian manifold. A Riemannian metric g on M tames h if g H = h. Natural question Is there a relation between the geodesics of g and the ones of h? Let V = H with respect to g. Define v = g V. Technical tool The following formula defines a conection (Bott) X Y =pr H g pr H X pr HY + pr V g pr V X pr VY + pr H [pr V X, pr H Y ] + pr V [pr H X, pr V Y ]. Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

20 Key Lemma Lemma (G., Grong) If {, } denotes the Poisson bracket wrt ω, then {H h, H v } = {H h, H g } = 0 g = 0 If Π M : T M M is the canonical projection, then exp sr : U x T x M M, exp r : V x T x M M, exp sr (x, tp) = (Π M e t H h )(p) exp r (x, t p) = (Π M e t H g )(p) Consequence exp r (x, t p) = ( Π M e t H h e t H v) (p) = ( Π M e t H v e t H h) (p). Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

21 Main result I Theorem (G., Grong) (M, g) Riemannian + F tot. geodesic Riem. foliation of M w/subbundle V. Define (M, H, h) where H = V and h = g H. Then, for any x M and p T M exp sr (x, tp) = exp r (exp r (x, t p), tpr V P t p), where P t is the parallel transport along exp r (x, t p) Compare with Montgomery s geodesics γ sr (t) = exp r (tv) exp G ( ta(v)). Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

22 Main result II If we assume that V is integrable, then we know more. Theorem (G., Grong) Every curve of the form exp sr (x, tp) is the horizontal lift of the projection of the curve exp r (x, t p) iff (a) V is the orthogonal complement of H. (b) The leaves of the foliation of V are totally geodesic. For the interested few: Riemannian and Sub-Riemannian Geodesic Flows. To appear J. Geom. Analysis (I guess this year) Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

23 Examples For principal bundles (+ technical conditions), the formula exp sr (x, tp) = exp r (exp r (x, t p), tpr V P t p) coincides with Montgomery s result The result can be applied to S 7 S 15 S 8, but we have no explicit formulas yet (M.Sc. problem anyone?) If (M, H, h), where H = ker α for α Ω 1 (M) contact, then g = 0 iff L Z g = 0, where Z is the Reeb vector field, g(z) = 1 and g H = h Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24

24 Mauricio Godoy M. (UFRO) Geodesic Flows February, / 24