Subriemannian geodesics - an introduction

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1 - an introduction Department of Theoretical Physics Comenius University Bratislava Student Colloqium and School on Mathematical Physics, Stará Lesná, Slovakia, August 24-30, an introduction

2 We will learn: Introduction What is subriemannian geometry - an introduction

3 We will learn: What is subriemannian geometry What are subriemannian geodesics - an introduction

4 We will learn: What is subriemannian geometry What are subriemannian geodesics How to set differential equations for finding them - an introduction

5 We will learn: What is subriemannian geometry What are subriemannian geodesics How to set differential equations for finding them Where subriemannian geodesics might be useful - an introduction

6 We will learn: What is subriemannian geometry What are subriemannian geodesics How to set differential equations for finding them Where subriemannian geodesics might be useful What are isoholonomic and (dual) isoperimetric problems - an introduction

7 We will learn: What is subriemannian geometry What are subriemannian geodesics How to set differential equations for finding them Where subriemannian geodesics might be useful What are isoholonomic and (dual) isoperimetric problems How they are related to subriemannian geodesics - an introduction

8 Contents Introduction 1 Introduction 2 Riemannian versus subriemannian Particular case - principal bundle with connection 3 Riemannian geodesics (review) 4 What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... - an introduction

9 Riemannian geometry (review) Riemannian versus subriemannian Particular case - principal bundle with connection In Riemannian geometry, at each x M, there exists g(u, v) g µν u µ v ν for each pair of vectors u, v (at the same point x M) - an introduction

10 Horizontal distribution (H, g) on M Riemannian versus subriemannian Particular case - principal bundle with connection In subriemannian geometry, at each x M there is, first, a distinguished k-dimensional subspace H x T x M called horizontal subspace in x. Now, the concept of the (positive definite) scalar product g(u, v) g µν u µ v ν is only defined for u, v from H x. - an introduction

11 Riemannian versus subriemannian Particular case - principal bundle with connection Horizontal distribution (H, g) on M (cont.) So, altogether, a k-dimensional horizontal distribution H on M is given the subriemannian metric, g, is (only) defined in H - an introduction

12 Subriemannian structure on M Riemannian versus subriemannian Particular case - principal bundle with connection The structure (M, H, g) is known as the subriemannian structure on M. It is studied in subriemannian geometry. Alternative names: Carnot-Carathéodory geometry (Gromov,...) Non-holonomic Riemannian geometry (Vershik-Gershkovich) Singular Riemannian geometry (Hermann,...) - an introduction

13 Riemannian versus subriemannian Particular case - principal bundle with connection How to describe a distribution on M Two most frequent ways how a distribution D is described: k (linearly independent) vector fields are given n k (linearly independent) covector fields are given In particular, let (e a, e i ) be a frame and (e a, e i ) the dual coframe. Then the prescriptions D = Span {e a } D = common null-space of e i = Ker e i define the same distribution. - an introduction

14 Riemannian versus subriemannian Particular case - principal bundle with connection How to describe a distribution on M (cont.) An alternative way how a distribution D may be described: let a ( 2 0) -tensor field h (of rank k) be given Then D = Im h (the range = image of h) provides a k-dimensional distribution, where h is regarded as a linear map h x : T x M T x M α h(α,. ) α µ α ν h νµ - an introduction

15 Riemannian versus subriemannian Particular case - principal bundle with connection How to describe a distribution on M (cont.) We want to "optimize" description of D in terms of h. Fix an adapted frame (e a, e i ), i.e. e a D. Write h = h ab e a e b + h ai e a e i + h ia e i e a + h ij e i e j i.e. Then ( h ab h h ai ) h ia h ij h(α,. ) = = (α b h ba + α i h ia )e a + (α a h ai + α j h ji )e i Since D = Span {e a }, we get h ai = 0 = h ji - an introduction

16 Riemannian versus subriemannian Particular case - principal bundle with connection How to describe a distribution on M (cont.) The simplest additional choice is h ia = 0, h ab = h ba So, finally, a distribution D = Span {e a } is encoded in h = h ab e a e b h ab = h ba with any symmetric, rank k matrix h ab. Or, equivalently, in ( h ab h h ai ) ( ) h ab 0 = 0 0 h ia h ij h ab = h ba - an introduction

17 H and g in a single object - the cometric h Riemannian versus subriemannian Particular case - principal bundle with connection Now we need to tell h about the metric g in H. Let g ab := g(e a, e b ). It is non-singular there exist (unique) inverse g ab. Define tensor g ab e a e b. Take this as h (i.e. choose h ab = g ab ) h = h ab e a e b h ab g bc = δ a c In a component-free language g(h(α), v) = α, v, v H Single tensor h, the cometric on M, carries full information about both D and g. is given by a pair (M, h). - an introduction

18 Riemannian geometry as a particular case Riemannian versus subriemannian Particular case - principal bundle with connection Riemannian geometry is a particular case, when H x = T x M (so that g operates on all pairs of vectors in x). Riemannian geometry is usually described by g = g ab e a e b metric but (since it is non-degenerate) one can equivalently use g = g ab e a e b cometric So the cometric is (in this sense!) more universal object, it works in both Riemannian and subriemannian case. - an introduction

19 Subriemannian hamiltonian Riemannian versus subriemannian Particular case - principal bundle with connection There is even a simpler object equivalent to h. Recall that we can canonically "lift" completely symmetric contravariant tensors on M into (particular) functions on T M. The general formula is t(p) := t(p,..., p) or, in canonical (Darboux) coordinates (x µ, p µ ) on T M t(x, p) := t µ...ν (x)p µ... p ν - an introduction

20 Subriemannian hamiltonian (cont.) Riemannian versus subriemannian Particular case - principal bundle with connection If, say, V = V µ (x) µ is a vector field on M, the corresponding function on T M V = V µ (x)p µ is called momentum corresponding to V. - an introduction

21 Subriemannian hamiltonian (cont.) Riemannian versus subriemannian Particular case - principal bundle with connection In particular, for the coordinate basis vector fields µ and the frame vector fields e a = e µ a (x) µ we get µ = p µ the µ-th canonical momentum e a = e a µ (x)p µ P a (x, p) the a-th canonical momentum - an introduction

22 Subriemannian hamiltonian (cont.) Riemannian versus subriemannian Particular case - principal bundle with connection And, finally, for the cometric h on M h = h ab e a e b we obtain the function on T M, the subriemannian Hamiltonian corresponding to h H(x, p) 1 2 h = 1 2 hab P a P b = 1 2 hab (x)(e a µ (x)p µ )(eb ν(x)p ν) 1 2 hµν (x)p µ p ν - an introduction

23 Total space as a subriemannian manifold Riemannian versus subriemannian Particular case - principal bundle with connection Let (M, g) be a Riemannian manifold. Let π : P M be a principle bundle with connection ω. Then, there is natural subriemannian structure on P. It is given in terms of the connection and the metric on M: Hor p := Ker ω p g p (u, v) := g x (π u, π v) - an introduction

24 Riemannian geodesics (review) Riemannian geodesics - Lagrangian language In the Riemannian case, on (M, g), the geodesic equations read ( γ γ) µ ẍ µ + Γ µ νρẋ ν ẋ ρ = 0 They are second order equations on M and turn out to coincide with the Euler-Lagrange equations for the (free motion) Lagrangian L(x, v) = 1 2 g µν(x)v µ v ν - an introduction

25 Riemannian geodesics (review) Riemannian geodesics - Hamiltonian language But L(x, v) = 1 2 g µν(x)v µ v ν H(x, p) = 1 2 g µν (x)p µ p ν so the geodesic equations may also be written as first order equations on T M in the form of the Hamiltonian equations ẋ µ = H/ p µ = g µν p ν ṗ µ = H/ x µ = (1/2)g νρ,µp ν p ρ - an introduction

26 Riemannian geodesics (review) Riemannian geodesics - Hamiltonian language (cont.) A useful observation: this particular Hamiltonian is nothing but the "lift" of the "inverse" metric tensor (cometric): if g 1 := g µν µ ν then H(x, p) 1 2 g µν (x)p µ p ν = 1 2 g 1 What s nice: this is a coordinate-free expression for "geodesic" Hamiltonian H(x, p) on T M out of g on M. - an introduction

27 Horizontal curves Introduction Riemannian geodesics (review) Whenever a distribution D is defined on a manifold M, there are specific curves γ(t) such that γ D They "reside in the distribution". In particular, for the horizontal distribution H on a subriemannian manifold (M, h) the curves are called horizontal curves. - an introduction

28 Riemannian geodesics (review) Horizontally lifted curves on the total space P Important example: let γ be a curve on M and let γ h be its horizontal lift onto the total space of a principal bundle π : P M with connection. Then, the lift is also a horizontal curve in the sense of the subriemannian structure on P. - an introduction

29 The length of a horizontal curve Riemannian geodesics (review) For a horizontal curve on (M, h), the length of the velocity vector γ := g( γ, γ) makes sense. Then, the "standard" definition l[γ] := γ dt makes sense, too. γ - an introduction

30 Riemannian geodesics (review) Example - the length of a horizontally lifted curve In particular, for π : P (M, g), the length of a horizontally lifted curve γ h is a well-defined concept and it is given as l[γ h ] = l[γ] where the latter is (as usual) l[γ] = tb t A This is simply because g( γ, γ)dt g p ( γ h, γ h ) := g π(p) (π γ h, π γ h ) = g x ( γ, γ) - an introduction

31 Extremal horizontal curves Riemannian geodesics (review) Subriemannian distance of two points A and B on M is d(a, B) := inf l[γ] over horizontal curves that connect A and B. If no such curve exists, we set d(a, B) =. (The existence depends on details of the distribution.) Subriemannian geodesic is the path which realizes the (finite) distance. - an introduction

32 How to find subriemannian geodesics Riemannian geodesics (review) To find geodesics in the particular case, in Riemannian geometry, we can write down Hamiltonian equations with the "cometric" Hamiltonian H(x, p) = 1 2 g µν (x)p µ p ν 1 2 g 1 Remarkable fact: this is also true in subriemannian case! - an introduction

33 Riemannian geodesics (review) How to find subriemannian geodesics (cont.) Namely, if we write down Hamiltonian equations ẋ µ = H/ p µ ṗ µ = H/ x µ with the subriemannian Hamiltonian H(x, p) = 1 2 hab P a P b = 1 2 hab (x)(e µ a (x)p µ )(e ν b (x)p ν) then the projection to M of a solution is 1. automatically horizontal (!) 2. extremal (!) - an introduction

34 Why this method works Riemannian geodesics (review) Let (e a, e i ) be any adapted frame (e a H, orthonormal) on a given subriemannian manifold (M, h). We promote (e a, e i ) to become orthonormal with respect to an auxiliary Riemannian metric, i.e. define G such that G := e a e a + e i e i So ( ) δab 0 G 0 δ ij G 1 ( ) δ ab 0 0 δ ij - an introduction

35 Why this method works (cont.) Riemannian geodesics (review) On the Riemannian manifold (M, G), the geodesics may be expressed in the standard Hamiltonian language on T M, the Hamiltonian being H(x, p) = 1 G 1 = (P ap a + P i P i ) The projections onto M of the solutions are, however, not horizontal, in general. (There is no reason for the shortest path with respect to G to be horizontal.) - an introduction

36 Why this method works (cont.) Riemannian geodesics (review) We can force the shortest curves to be "willingly" horizontal by economical means - by severe penalization of "bad" (nonhorizontal) motions. Introduce new metric as ( ) δab 0 G λ 0 λ 2 δ ij G 1 λ with λ 2 >> 1 (still with respect to (e a, e i )). ( δ ab 0 ) 0 1 δ ij λ 2 - an introduction

37 Why this method works (cont.) Riemannian geodesics (review) If γ(t) is not horizontal somewhere, the segment of the curve around this point adds a huge contribution. So, the curves which are everywhere horizontal, are the only candidates to be reasonably short. Then, in the limit λ 2, the shortest paths, the solutions of Hamiltonian equations with H λ = 1 2 (P ap a + 1 λ 2 P ip i ) are automatically horizontal. - an introduction

38 Why this method works (cont.) Riemannian geodesics (review) But, in the limit λ 2, H λ = 1 2 (P ap a + 1 λ 2 P ip i ) 1 2 P ap a which is nothing but the subriemannian Hamiltonian corresponding to (M, h). - an introduction

39 What are isoholonomic problems What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading Isoholonomic means that the holonomy of a loop γ(0) = γ(1) is kept constant. And something else (namely the length) is to be minimized. So one tries to get a given holonomy via the shortest loop. - an introduction

40 What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading What are (dual) isoperimetric problems Isoperimetric means that a perimeter is kept constant. And something else (namely area) is to be maximized. The dual problem: keep the area constant and minimize the length. Isoperimetric classics - the Dido s problem: Find the shape of a strip (realized by an ox hide, originally) enclosing the maximum area, given the length of the strip constant. Solution: arc of a circle, see Prof. Virgil s Aeneid. - an introduction

41 What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading How (dual) isoperimetric becomes isoholonomic Let E 2 be the standard Euclidean plane. Construct a principal bundle π : P E 2 with the group G = (R, +) or U(1), such that F da = π ds where ds is the area 2-form in E 2. Then, for any section σ, σ F = (π σ) ds = ds - an introduction

42 What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading How (dual) isoperimetric becomes isoholonomic (cont.) Now, consider a loop γ(0) = γ(1) which happens to be the boundary of S γ = S Then, we get area of S := S ds = S σ F = γ σ A = holonomy of γ - an introduction

43 Ant on a turntable Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading Consider a gramophone turntable and an Ant living on it. When he performs a loop-shaped walk, at the end the turntable gets rotated. This can be treated in terms of a SO(2)-bundle with connection over the turntable: the net rotation is the holonomy of the loop. So, we come to a subriemannian structure discussed above. - an introduction

44 What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading Ant on a turntable - resulting Hamiltonian equations ṙ = p r ϕ = pϕ α = ṗ r = ṗ ϕ = 0 ṗ α = 0 ( pα r 2 1+r 2 pϕ pα ( r 2 pϕ r 2 1+r 2 ) ( ) ( ϕ 1+r 2 r 2 pα 1+r 2 ) ( 2rpα 1+r 2 ) ϕ ) r 2 1+r 2 ( ) 2rpα 1+r 2 - an introduction

45 Falling cat Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading Old wisdom: a cat, dropped from upside down, will land on her feet. Freely falling reference frame - no fall. So she is able to reorient herself while floating weightless in space. Physical mechanism: angular momentum equals zero. Mathematical expression: there is an SO(3)-principal bundle with connection and the resulting net rotation is the holonomy for the loop in the space of abstract shapes. picture from Montgomery - an introduction

46 Berry phase Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading In Quantum Mechanics of (n + 1)-level systems, there is a natural U(1)-principal bundle π : S 2n+1 CP n Hopf bundle, if n = 1 Here, S 2n+1 is the space of normed vectors in the Hilbert space C n+1 whereas CP n is the space of the states. There is a distinguished metric tensor on CP n (Fubiny-Study). In addition, there is also a canonical connection in the bundle. So, we come to a subriemannian structure discussed above. - an introduction

47 For Further Reading Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading R. Strichartz. Sub-Riemannian Geometry. J.Diff.Geom. 24, , an introduction

48 For Further Reading Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading R. Strichartz. Sub-Riemannian Geometry. J.Diff.Geom. 24, , R. Montgomery. Isoholonomic Problems and some Applications. Commun.Math.Phys., 128, , an introduction

49 For Further Reading Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading R. Strichartz. Sub-Riemannian Geometry. J.Diff.Geom. 24, , R. Montgomery. Isoholonomic Problems and some Applications. Commun.Math.Phys., 128, , R. Montgomery. A Tour of Subriemannian Geometries, Their Geodesics and Applications. American Mathematical Society, an introduction

50 For Further Reading Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading R. Strichartz. Sub-Riemannian Geometry. J.Diff.Geom. 24, , R. Montgomery. Isoholonomic Problems and some Applications. Commun.Math.Phys., 128, , R. Montgomery. A Tour of Subriemannian Geometries, Their Geodesics and Applications. American Mathematical Society, Numerous other authors. Plenty of other papers and several monographs. Lot of other journals. - an introduction

51 For Further Reading Introduction What it is about Isoperimetric problems as a special case Ant on a turntable, falling cat, Berry phase,... Further Reading R. Strichartz. Sub-Riemannian Geometry. J.Diff.Geom. 24, , R. Montgomery. Isoholonomic Problems and some Applications. Commun.Math.Phys., 128, , R. Montgomery. A Tour of Subriemannian Geometries, Their Geodesics and Applications. American Mathematical Society, Numerous other authors. Plenty of other papers and several monographs. Lot of other journals. - an introduction

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