Subriemannian Geometry: The basic notations and examples Winterschool in Geilo, Norway Wolfram Bauer Leibniz U. Hannover March 4-10. 2018 W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 1 / 33 Outline 1. Motivations, definitions and examples 2. Horizontal curves and an optimal control problem 3. More examples and constructions W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 2 / 33
Motivation: Sub-Riemannian geometry � � Consider n classical particles with coordinates q 1 , · · · , q n . Motion under constraints H: f ( q 1 , · · · , q n ) = 0, (holonomic) , NH: f ( q 1 , · · · , q n , ˙ q 1 , · · · , ˙ q n ) = 0, (non-holonomic) . Exampels: H: A particle moving along a surface, or a pendulum. NH: Rolling of a ball on a plane (or some surface) without slipping or twisting. Corresponding geometric structures on a manifold holonomic constraints − → integrable distribution (foliation of a manifold), non-holonomic constraints − → Sub-Riemannian structure. W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 3 / 33 Parking a car: Rototranslation Position of the car robot in 3-space: ( x , y , ϑ ) ∈ R 2 × S 1 . Possible movements X = cos ϑ · ∂ x + sin ϑ · ∂ y , (in direction of the car) Y = ∂ ϑ , (rotation) Z = − sin ϑ · ∂ x + cos ϑ · ∂ y , (orthogonal to the car). W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 4 / 33
Parkin a car: Rototranslation Connecting positions : Which movements allow to reach from any position of the car any other position? Observations Moving only along X and Z is not enough: it keeps the angle ϑ fixed. � � span X , Z = kern d ϑ and d ϑ = closed form , [ X , Z ] = 0 . Moving along X and Y (parking procedure) might be sufficient for connecting positions. � � span X , Y = kern ω where ω = − sin ϑ dx + cos ϑ dy . � � [ X , Y ] = cos ϑ · ∂ x + sin ϑ · ∂ y , ∂ ϑ = − sin ϑ · ∂ x + cos ϑ · ∂ y = Z . W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 5 / 33 Sub-Riemannian Geometry ”Sub-Riemannian geometry models motions under non-holonomic constraints”. Definition A Sub-Riemannian manifold (shortly: SR-m) is a triple ( M , H , �· , ·� ) with: M is a smooth manifold (without boundary), dim M ≥ 3 and H ⊂ TM is a vector distribution. H is bracket generating of rank k < dim M , i.e. Lie x H = T x M . �· , ·� x is a smoothly varying family of inner products on H x for x ∈ M . W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 6 / 33
1.Example: Heisenberg group Consider the 3- dimensional Heisenberg group H 3 ∼ = ( R 3 , ∗ ) with product: � x 1 + x 2 , y 1 + y 2 , z 1 + z 2 + 1 � � � � � x 1 , y 1 , z 1 ∗ x 2 , y 2 , z 2 = 2[ x 1 y 2 − y 1 x 2 ] . Lie algebra of H 3 : = R 3 define left-invariant vector fields: Let q = ( x , y , z ) ∈ H 3 : 1 On H 3 ∼ � � ( q ) = df � � X 1 f q ∗ ( t , 0 , 0) dt | t =0 �� ∂ � � = df � x + t , 0 , z − yt � ∂ x − y ∂ = f ( q ) . dt 2 2 ∂ z Similarly, with curves (0 , t , 0) t and (0 , 0 , t ) t : X 2 = ∂ ∂ y + x ∂ Z = ∂ and ∂ z . 2 ∂ z 1 ” X left-invariant”: X g ∗ h = ( L g ) ∗ X h with the left-multiplication L g : H 3 → H 3 . W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 7 / 33 Heisenberg group as SR-manifold Known fact: The Lie algebra ( h 3 , [ · , · ]) of H 3 can be identified with: � � h 3 = span X 1 , X 2 , Z with [ · , · ] = commtator of vector fields . Observation If we calculate Lie-brackets [ · , · ], then one only finds one non-trivial bracket relation is: � � X 1 , X 2 = X 1 X 2 − X 2 X 1 = Z . Put H = span { X 1 , X 2 } ⊂ T H 3 (distribution), Define �· , ·� on H by declaring X 1 and X 2 pointwise orthonormal. Conclusion: ( H 3 , H , �· , ·� ) defines a Sub-Riemannian structure on H 3 . W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 8 / 33
Horizontal curves and cc-distance: On a SR-manifold ( M , H , �· , ·� ) we consider horizontal objects, i.e. objects under non-holonomic constraints. Example Consider a curve γ : [0 , 1] → M : a γ is called horizontal, (a.e.) it is tangential to H , i.e. γ ( t ) ∈ H γ ( t ) . ˙ The curve length is defined by: � 1 �� � ℓ ( γ ) := γ ( t ) , ˙ ˙ γ ( t ) γ ( t ) dt . 0 SR geodesic = locally length minimizing horizontale curve. a piecewise C 1 or just absolutely continuous W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 9 / 33 Carnot-Carath´ eodory metric Definition: Sub-Riemannian distanced (cc-distance) The SR distance between two points a , b ∈ M is defined by: � � d cc ( a , b ) := inf ℓ ( γ ) : γ horizontal , γ (0) = a , γ (1) = b . Question: Let M be a connected SR-manifold. Can we connect any two points on M by horizontal curves? Theorem (W.-L. Chow 1939, P.-K. Rashevskii 1938) Any two points on a connected SR-manifold can be connected by piecewise smooth horizontal curves. Consequence: The cc-distance d cc 2 on a connected SR-manifold is finite. Hence ( M , d cc ) forms a metric space. 2 Carnot-Carath´ eodory distance W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 10 / 33
Geodesic equations Some question: How can we obtain Sub-Riemannian geodesics? Relation to d cc : can we realize the cc-distance between two point by a (piecewise) smooth SR geodesic? Is the distance x �→ d cc ( x 0 , x ) smooth for fixed points x 0 ? Let ( M , H , �· , ·� ) be a SR-manifold. Let � � X 1 , · · · , X m = vector fields and m = rank H . an local orthonormal frame around a point q ∈ M , i.e. � � � � H q = span X 1 ( q ) , · · · , X m ( q ) and X i ( q ) , X j ( q ) = δ ij . Idea: Expand locally the derivative of a horizontal curve with respect to the above frame W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 11 / 33 SR-geodesics and optimal control Observation Let γ : [0 , 1] → M be horizontal. With suitable coefficients u i ( t ) one can write m m � � � � γ ′ ( t ) = γ ′ ( t ) , γ ′ ( t ) u 2 u j ( t ) · X j ( t ) = ⇒ = i ( t ) . j =1 j =1 Finding SR-geodesics between A , B ∈ M = optimal control problem OCP. OCP : Minimize the cost � � T m � J T ( u ) := 1 � � u 2 i ( t ) dt � 2 0 j =1 under the conditions m � γ ′ = u j · X j ( γ ) and γ (0) = A , γ ( T ) = B . j =1 W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 12 / 33
SR-geodesic: a Hamiltonian formalism Remark : Instead of minimizing a lenght we may equivalently minimize an ”energy” : OCP : Minimize the cost � T m J T ( u ) := 1 � u 2 i ( t ) dt 2 0 j =1 under the conditions m � γ ′ = u j · X j ( γ ) and γ (0) = A , γ ( T ) = B . j =1 Hamiltonian formalism (as known in Riemannian geometry): Assign a Sub-Riemannian Hamiltonian H sr ∈ C ∞ ( T ∗ M ) to the problem: m � � 2 � ( q , p ) ∈ T ∗ H sr ( q , p ) = p X j ( q ) where q M . j =1 W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 13 / 33 SR-geodesic: a Hamiltonian formalism With the Poisson bracket {· , ·} on C ∞ ( T ∗ M ) consider: → = ∂ H ∂ p · ∂ ∂ q − ∂ H ∂ q · ∂ � � H sr = · , H ∂ p = Hamiltonian vector field The Hamiltonian vector field defines the geodesic flow on T ∗ M and projections of the flow to M give SR-geodesics: Theorem (normal geodesics) Let ζ ( t ) = ( γ ( t ) , p ( t )) be a solution to the normal geodesic equations: q = ∂ H p = − ∂ H ˙ ( q , p ) and ˙ ( q , p ) , i = 1 · · · dim M . ∂ p i ∂ q i Then γ ( t ) locally minimizes the SR-distance. Proof: 3 3 R. Montgomery, A tour of Subriemannian Geometries, Their Geodesics and Applications Math. Surveys and Monographs, 2002. W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 14 / 33
SR-geodesics Remark There are various differences to the setting of a Riemannian manifold: The Hamiltonian in Riemannian geometry can be expressed as n � g ij := inverse metric tensor. g ij ( q ) p i p j , H r ( q , p ) = i , j =1 In SR-geometry g ij is an m × m -matrix and not invertible on TM . There are no 2nd order geodesic equations in the SR-setting such as q k = Γ k ¨ ij ˙ q i ˙ q j . The obtained regularity of SR-geodesics is not clear. In SR-geometry there may be singular geodesics which do not solve the geodesic equations in the above theorem. W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 15 / 33 The falling cat : A connectivity problem in SR geometry W. Bauer (Leibniz U. Hannover ) Subriemannian geometry March 4-10. 2018 16 / 33
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