Some topics in sub-riemannian geometry

Transcription

1 Some topics in sub-riemannian geometry Luca Rizzi CNRS, Institut Fourier Mathematical Colloquium Universität Bern - December

2 Sub-Riemannian geometry Known under many names: Carnot-Carathéodory geometry (Gromov, 1981) Singular Riemannian geometry (Brockett, 1981) Sub-Riemannian geometry (Strichartz, 1986) Non-holonomic geometry (Russian school, from mechanics) Sub-elliptic geometry (from PDEs) Riemannian geometry with non-holonomic constraints Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 1 / 26

3 Dido s isoperimetric problem Given a string of fixed length l and a fixed line L (the Mediterranean coastline), place the ends of the string on L and determine the shape of the curve c for which the figure enclosed by c together with L has the maximum possible area. Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 2 / 26

4 Dido s dual problem Dido s dual problem Minimize the length of a curve with given endpoints such that the enclosed area is fixed Ω c(t) = (x(t), y(t)) L(c) = 1 0 ẋ(t) 2 + ẏ(t) 2 dt A(Ω) = dx dy = 1 d (xdy ydx) = 1 (xdy ydx) Ω 2 Ω 2 c Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 3 / 26

5 Equivalent formulation of Dido s dual problem z (x(t), y(t), z(t)) γ x Lift c R 2 γ R 3 such that z = A(Ω) Lift obeys non-holonomic constraint y c Ω ω( γ) = 0, ω := dz 1 (xdy ydz) 2 Reformulation of Dido s dual problem Find γ such that ω( γ) = 0, minimizing L(c) and with fixed endpoints. Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 4 / 26

6 Towards a formal definition Definition A sub-riemannian structure on a smooth manifold M is a pair (D, g) D T M is a smooth vector distribution g is a smooth scalar product on D Admissible (or horizontal) curves γ : [0, 1] M such that γ D γ(t) L(γ) := 1 Dido s dual problem (Heisenberg group) M = R 3 0 γ(t) g dt D = ker ω, with ω = dz 1 2 (xdy ydx) g is the euclidean product of the projection on R 2 admissible curves are all the lifts Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 5 / 26

7 The problem of accessibility Lie bracket: Let X, Y be vector fields and φ X t the flow φ Y t φ X t φ Y t φ X t (x) = x + t 2 [X, Y ](x) + o(t 2 ) φ X t φ Y t φ Y t t 2 [X, Y ] x φ X t Theorem (Frobenius, 1877) If [D, D] D, then M is foliated by maximal integral submanifolds of D. If D is integrable no connectedness by admissible path (even locally) Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 6 / 26

8 Chow-Rashevsky Theorem Not-integrable is not enough for connectedness, even in corank 1 Definition (Hörmander condition) D = span{x 1,..., X k } is completely non-integrable if span{[x i1, [X i2,..., [X ij 1, X ij ]] x j 1} = T x M, x M completely non-holonomic, Lie-bracket generating Theorem (Chow 39, Rashevsky 38) If D is completely non-integrable, then for all x M there exists a neighborhood U such that, for all points y, z M there exists an admissible curve joining them. If M is connected M is admissible-path connected Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 7 / 26

9 Example: car robot Position + orientation (x, y, θ) R 2 S 1 X θ (x, y) X = cos θ x + sin θ y Y = θ Z = sin θ x + cos θ y (move forward) (steer) (move orthogonally) Admissible directions: D = span{x, Y } verifies [X, Y ] = Z! Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 8 / 26

10 Sub-Riemannian manifolds Definition A sub-riemannian structure on a smooth manifold M is a pair (D, g) D T M is a vector distribution D satisfies the Hörmander condition g is a smooth scalar product on D Remark: Riemannian Sub-Riemannian Sub-Riemannian metric structure d SR (x, y) := inf{l(γ) γ admissible, γ(0) = x, γ(1) = y} By Chow-Rashevsky, d SR : M M R is finite continuous (metric topology = topology of M) Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 9 / 26

11 Hypoelliptic operators Let µ be a smooth measure on M. The sub-laplacian is where X i is a local o.n. frame, X i Theorem (Hörmander 1967) k µ := Xi X i i=1 is the formal adjoint in L 2 (M, dµ). Assume {X 1,..., X k } satisfy Hörmander condition. Then µ f = u, u C c (M) has a unique smooth solution f of compact support. Plays the role of Laplace-Beltrami There is no intrinsic smooth measure in sub-riemannian geometry The principal symbol is always the same for all µ Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 10 / 26

12 Heisenberg group The simplest sub-riemannian manifold: M = R 3, with (D, g) generated by X 1 = x, X 2 = y + x z Length-minimizing curves starting from the origin solve dual Dido Figure: Heisenberg distribution Figure: Heisenberg unit sphere Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 11 / 26

13 Sub-Riemannian balls Flag of the distribution at x D 1 x D 2 x D m x T x M, D i+1 := D i + [D, D i ] Let d i = dim Dx i dim Dx i 1. The non-holonomic box at x is Box x (ε) := [ ε, ε] d 1 [ ε 2, ε 2 ] d 2 [ ε m, ε m ] dm Theorem (Ball-Box) Let x M. There exist coordinates, c < C and ε 0 > 0 such that Box x (cε) Ball SR (x, ε) Box x (Cε), ε ε 0 Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 12 / 26

14 Hausdorff dimension Dimension defined for any metric space (X, d) Rough definition The Hausdorff dimension of A X is Q if the minimal number N(ε) of balls of radius ε required to cover A satisfies N(ε) ε Q for ε 0. Figure: Hausdorff dimension for the west coast of GB 1.25 Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 13 / 26

15 Mitchell formula Theorem (Ball-Box for the Heisenberg group) There exist c < C such that, for all ε > 0, Box(cε) Ball SR (ε) Box(Cε), where Box(ε) = [ ε, ε] [ ε, ε] [ ε 2, ε 2 ] To cover A H 3 optimally, I need N(ε) ε 4 balls of radius ε Hausdorff dimension of Heisenberg group = 4 Theorem (Mitchell 1985) For a (equiregular) sub-riemannian structure of step m m Hausdorff dimension = i(rank D i rank D i 1 ) i=1 Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 14 / 26

16 Topology of small balls Open Problem Are small sub-riemannian balls homeomorphic to euclidean balls? Affirmative answer for Carnot groups and step 2 (Baryshnikov, 2000). Figure: Martinet Figure: Heisenberg Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 15 / 26

17 Sub-Riemannian geodesics Geodesic problem (equivalent formulations) Fix (D, g) on M. For x, y M, find admissible curves γ : [0, 1] M, that minimize the length (equivalently, the energy functional) L(γ) = 1 0 γ(t) dt, ( J(γ) = 1 1 ) γ(t) 2 dt 2 0 Remark: Minimizers of J are parametrized with constant speed and coincide with (constant speed reparametrizations of) minimizers of L Riemannian case: Levi-Civita connection + variation formulas = D t γ = 0 ẍ + Γ k ijẋ i ẋ j = 0 Riemannian geodesics are determined uniquely by their initial velocity SR case: too rich techniques from optimal control theory Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 16 / 26

18 Sub-Riemannian geodesics Assume D = span{x 1,..., X k } (global orthonormal frame) A curve is admissible if there exists a control u such that k ( ) γ(t) = u i (t)x i (γ(t)), u L 2 ([0, 1], R k ) i=1 Definition (End-point map) Let γ u (1) be the end-point of the solution of ( ) starting at γ u (0) = x. E x : L 2 ([0, 1], R k ) M, E x (u) := γ u (1) It is a smooth, open map. Geodesic problem = constrained minimum problem: J(u) = u(t) 2 dt min, E x (u) = y Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 17 / 26

19 Geometric Lagrange multipliers rule J(u) = γ u minimizer two possibilities: 0 u(t) 2 dt min, u Ex 1 (y) 1 y is a regular value of E x (standard Lagrange multipliers rule) 2 y is a critical value of E x, i.e. d u E x is not surjective Necessary condition for minimality (Lagrange multipliers rule) Let γ u be a minimizer of J. Then there exists λ T y M, ν {0, 1} s.t. νd u J λ d u E x = 0, (λ, ν) 0 1 ν = 1 λ is normal multiplier and γ a normal extremal 2 ν = 0 λ is abnormal multiplier and γ an abnormal extremal Remark: (ν, λ) is not unique, so γ can be normal and abnormal Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 18 / 26

20 Normal extremals Sub-Riemannian Hamiltonian H : T M R H(p, x) = 1 k (p X i (x)) 2, H H = 2 p i=1 x H x p Normal extremals = projections of solutions of Hamilton s equations T M λ 0 H λ(t) Fiber-wise quadratic form n H(p, x) = 1 2 i,j=1 g ij (x)p i p j x γ(t) π y M Degenerate Determined by their initial covector! Locally minimizing Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 19 / 26

21 Abnormal extremals Abnormal extremal: curve γ u that satisfies the abnormal multiplier rule: λ d u E x = 0, 0 λ T y M = u Crit(E x ) Riemannian case: E x is always a submersion no abnormal curves Open Problem (SR Sard conjecture) What is the size of E x (Crit(E x ))? Zero measure? Empty interior? Recent results by Rifford, Belotto and Le Donne, Montgomery, Ottazzi, Pansu, Vittone Theorem (Sard Theorem) Let f : M N be a C r map. Then if k > max{0, m n}, the set of critical values of f has zero measure Fails in infinite dimensions! Take f : l 2 R f({x n }) = (3 2 n/3 x 2 n 2x 3 n), f(crit(f)) = [0, 1] n=1 Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 20 / 26

22 Regularity of geodesics Open Problem (Regularity of minimizers) Are all sub-riemannian minimizing curves smooth? Various false proofs that all minimizers are normal Montgomery, 1991: strictly abnormal minimizers do exist! (smooth) One can build non-smooth abnormal curves (are they minimizing?) Yes for step 2, and generic rank 3 (Chitour, Jean, Trélat 2006) Theorem (Hakavouri - Le Donne 2016) Length-minimizing curves do not have corners It relies on previous results of Leonardi, Monti, Vittone Theorem (Sussman 2014) In the analytic setting, length-minimizing curves are analytic on an open dense subset of their domain Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 21 / 26

23 Why Carnot-Carathéodory geometry? Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 22 / 26

24 Why Carnot-Carathéodory geometry? Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 23 / 26

25 Thermodynamics revisited M = space of thermodynamic states of closed system curve γ in M = thermodynamic (reversible) transformation Not all transformations are allowed non-holonomic constraint First Law of Thermodynamics There exists a function U and one-forms δq, δw, such that du = δq + δw U is the energy of the system, δq measures the heat exchange and δw the work done on the system. Adiabatic transformation: γ ker(δq) (a corank 1 distribution) No work transformation: γ ker(δw ) (a corank 1 distribution) Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 24 / 26

26 Axiom II (Charathéodory 1909) In an arbitrary neighborhood of an arbitrarily given initial state of a system (of any number of thermodynamic coordinates) there is a state that is inaccessible by reversible adiabatic process. (locally the space of states is not connected by adiabatic curves) Theorem (on Pfaff equations, Charathéodory, 1909) Given a Pfaff equation δq = dx 0 + X 1 dx X n dx n = 0 with X i differentiable, such that in any neighborhood of any point x there are points that are not path connected to x by curves γ satisfying δq( γ) = 0, then δq = T ds for some T > 0 and S. For adiabatic processes δq = T ds (existence of entropy) Same principle for no work processes δw = pdv Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 25 / 26

27 Back to contact geometry First Law of Thermodynamics For all thermodynamic transformations γ : [0, 1] M du δq δw = 0 Carathéodory Theorem on Pfaff equations δq = T ds, δw = pdv First Law + Axiom II, thermodynamic transformation γ ω := du T ds + pdv, ω( γ) = 0 ω dω dω = du dt ds dp dv 0 Hörmander condition! Thermodynamic transformations are admissible curves of a contact distribution (Legendrian curves) Luca Rizzi (CNRS, Institut Fourier) Some topics in sub-riemannian geometry 26 / 26