Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian Irina Markina, Der Chen Chang, and Alexander Vasiliev University of Bergen, Norway Georgetown University, USA Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 1/50
Definitions Let M n be a differentiable manifold, TM n be a tangent bundle, and �· , ·� be a positively definite metric on TM n ( M n , TM n , �· , ·� TM n ) is a Riemannian manifold Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 2/50
Definitions Let M n be a differentiable manifold, TM n be a tangent bundle, and �· , ·� be a positively definite metric on TM n ( M n , TM n , �· , ·� TM n ) is a Riemannian manifold Take a manifold M n , a distribution of k -dimensional planes D k ⊂ TM n , k < n , and a positively definite metric �· , ·� on D k ( M n , D k , �· , ·� D k ) is a sub-Riemannian manifold Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 2/50
Examples • Parallel parking, Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 3/50
Examples • Rolling ball without slipping and twisting, Sub-Riemannin manifold: ( R 5 , R 2 , Euclidean metric on the plane ) Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 4/50
Examples • falling cat How does a cat falling in mid-air with no angular momentum, spin itself around and right itself? Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 5/50
Examples • swimming Microorganisms live in an environment dominated by viscous drag and Brownian motion. How a cyclic motion of a body results to propel it forward? Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 6/50
Geometry of principal bundles Let M be a configuration space of a mechanical system with a kinetic energy T and a potential energy U . If a group G acts on M : G � M freely and leaves the energies invariant, then the quotient map h : M → M/G = Q gives the configuration space M the structure of the principal G -bundle. We pullback the metric ρ Q to M : h ∗ ρ Q ( X, Y ) = ρ Q ( h ∗ X, h ∗ Y ) and it gives the sub-Riemannian structure to M : ( M, TQ, h ∗ ρ Q ) . Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 7/50
Geometry of principal bundles To the shortest curve γ in the configuration space M corresponds the shortest curve c in the base space Q (which is the projection under h : M → Q ). Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 8/50
Falling cat Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 9/50
Falling cat M is a space of "shape of the cat" together with "her orientation" and "location in the space". The group G = SE (3) is the group of rigid motions that actually can be reduced to SO (3) . Thus Q = M/G is a space of pure shapes. Since the initial shape is the same as the final, the problem to find the optimal way of falling is to find the shortest loop in the space of pure shapes Q . Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 9/50
Connection ( M n , D k , �· , ·� D k ) is a sub-Riemannian manifold Can we join any points by a curve? Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 10/50
Bracket generating condition A distribution D k = span { X 1 , . . . , X k } is called bracket generating if X 1 , . . . , X k together with all of its iterated Lie brackets [ X i , X j ] , [ X i , [ X j , X m ]] , . . . span the tangent bundle TM n . If D k is bracket generating and M n is connected, then any two points can be connected by a horizontal curve γ ∈ D k . ˙ Chow-Rashevskii theorem (1938-1939) 1. W. L. Chow : Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. , 117 (1939), 98-105. 2. P. K. Rashevskii : About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski ped. inst. Libknekhta , 2 (1938), 83-94. in Russian Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 11/50
Carnot-Carathéodory distance Given bracket generating D k and connected M n there is a γ (0) = x, γ (1) = y, ∀ x, y ∈ M n γ : [0 , 1] → M : k � such that γ ( t ) = ˙ α i ( t ) X i ( γ ( t )) i =1 The Carnot-Carathéodory distance is � 1 γ ( t ) � 1 / 2 d c − c ( x, y ) = inf { � ˙ γ ( t ) , ˙ D k dt : 0 γ is horizontal and connects the points x and y } . Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 12/50
Example. Heisenberg group X = ∂ x − 1 Y = ∂ y + 1 R 3 , 2 y ∂ z , 2 x ∂ z , Z = [ X, Y ] = XY − Y X = ∂ z , ds 2 = dx 2 + dy 2 span { X, Y, [ X, Y ] } = R 3 , ( R 3 , R 2 = span { X, Y } , ds 2 ) is the Heisenberg group ( R 3 , +) ( R 3 , ◦ ) ⇒ Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 13/50
Horizontal curve on Heisenberg group z + 1 � � γ = ˙ ˙ x ∂ x + ˙ y ∂ y + ˙ z ∂ z = ˙ xX + ˙ yY + ˙ 2( y ˙ x − x ˙ y ) Z z = 1 The horizontality condition ˙ 2 ( x ˙ y − y ˙ x ) Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 14/50
Geodesics Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 15/50
Geodesics Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 16/50
Geodesics ω = dz − 1 2 ( x dy − y dx ) , Ω = dω = dx ∧ dy , d Ω = 0 d − → v dt = − → − → v × Ω , v = ( ˙ x, ˙ y ) , Ω = 0 dx ∧ dz + 0 dz ∧ dy + 1 dx ∧ dy Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 17/50
Heisenberg ball Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 18/50
SU (2) group � � z 1 z 2 | z 1 | 2 + | z 2 | 2 = 1 . SU (2) : , z 1 , z 2 ∈ C , − ¯ z 2 z 1 ¯ ( z 1 , z 2 ) − 1 = (¯ (1 , 0) is the unit z 1 , − z 2 ) , S 3 = { x ∈ R 4 : x 2 1 + x 2 2 + x 2 3 + x 2 for 4 = 1 } z 1 = x 1 + ix 2 , z 2 = x 3 + ix 4 U (1 , H ) is the group of unit quaternions, Sp (1) is the special symplectic group, Spin (3) is the spin group on three generators. And they are double cover of the group SO (3) . Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 19/50
Left invariant vector fields x 1 − x 2 − x 3 − x 4 1 x 1 x 2 x 1 − x 4 x 3 0 x 2 � � L q ( · ) ∗ = = x 3 x 4 x 1 − x 2 0 x 3 x 4 − x 3 x 2 x 1 0 x 4 N = x 1 ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 + x 4 ∂ 4 is normal vector to S 3 , � N, N � = 1 , Z = − x 2 ∂ 1 + x 1 ∂ 2 + x 4 ∂ 3 − x 3 ∂ 4 , � Z, Z � = 1 , X = − x 3 ∂ 1 − x 4 ∂ 2 + x 1 ∂ 3 + x 2 ∂ 4 , � X, X � = 1 , Y = − x 4 ∂ 1 + x 3 ∂ 2 − x 2 ∂ 3 + x 1 ∂ 4 , � Y, Y � = 1 , [ Z, X ] = 2 Y , [ Y, Z ] = 2 X , [ X, Y ] = 2 Z Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 20/50
Horizontality condition T p ( S 3 ) = span { X, Y, Z } , D = span { X, Y } . The geometry obtained by fixing other pair of vector fields is similar. Let γ ( s ) = ( x 1 ( s ) , x 2 ( s ) , x 3 ( s ) , x 4 ( s )) be a curve on S 3 . Then γ ˙ = x 1 ∂ 1 + ˙ ˙ x 2 ∂ 2 + ˙ x 3 ∂ 3 + ˙ x 4 ∂ 4 = a ( s ) X ( γ ( s )) + b ( s ) Y ( γ ( s )) + c ( s ) Z ( γ ( s )) . The curve γ is horizontal iff c = � ˙ γ, Z � = − x 2 ˙ x 1 + x 1 ˙ x 2 + x 4 ˙ x 3 − x 3 ˙ x 4 = 0 . The set of horizontal curves is not empty. Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 21/50
Horizontality condition 1 2( x 1 dx 2 − x 2 dx 1 ) = 1 2( x 3 dx 4 − x 4 dx 3 ) Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 22/50
Hopf fibration S 2 , h − 1 ( p ) = S 1 , p ∈ S 2 is the principal S 1 h : S 3 → bundle h ( z 1 , z 2 ) = ( | z 1 | 2 − | z 2 | 2 , 2 z 1 ¯ z 2 ) ∈ S 2 . It is submersion of S 3 onto S 2 and bijection between S 3 /S 1 and S 2 . The action of S 1 on S 3 is defined by e 2 πit · ( z 1 , z 2 ) = ( e 2 πit z 1 , e 2 πit z 2 ) , e 2 πit ∈ S 1 , ( z 1 , z 2 ) ∈ S 3 φ ( t ) = e 2 πit · (ˆ z 2 ) is a fiber over (ˆ z 2 ) that collapses z 1 , ˆ z 1 , ˆ to h (ˆ z 2 ) under the Hopf map. The curve φ ( t ) is a z 1 , ˆ great circle on S 3 . Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 23/50
Ehresmann connection d φ ( t ) h ( ˙ φ ( t )) = 0 ker( dh ) = span { ˙ φ ( t ) } = span { 2 πZ } ⊂ TS 3 The orthogonal complement to ker( dh ) is D = span { X, Y } : ker( dh ) ⊕ D = TS 3 The distribution D constructed in this way is called the Ehresmann connection. The metric on D coincides with the pull back of the metric on S 2 by the Hopf map. Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 24/50
Geodesics � ( a ( s ) 2 + b ( s ) 2 ) ds + λ ( s ) c ( s ) The geodesics are characterized by the following. The angle ∠ (˙ γ, X ( c ( s ))) increases linearly. X X Y X Sub-Riemannian view on SU (2) and semigroup of its sub-Laplacian – p. 25/50
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