Isometries of subRiemannian manifolds Alessandro Ottazzi (in collaboration with Enrico Le Donne) Alba, 18 giugno 2013 Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 1 / 14
Plan of the talk Notation and Setting 1 State of the art 2 Main results 3 Open questions 4 Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 2 / 14
subRiemannian manifolds A subRiemannian structure is a triple ( M , H , g ) , where M is a differentiable and connected manifold of dimension n ; H ⊆ TM is a subbundle of the tangent bundle TM with constant rank m ≤ n , that depends smoothly on p ∈ M and that satisfies H¨ ormander condition: span { [ X 1 , [ . . . , [ X i − 1 , X i ] . . . ]]( p ) | X i ∈ Γ ( H )] } = T p M for every p ∈ M . We call H the horizontal bundle. g is a Riemannian metric on H . If n = m , then M is a Riemannian manifold. Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 3 / 14
Regular subRiemannian manifolds H 1 : = H and H i + 1 : = H i + [ H i , H ] , where H i + 1 ( p ) : = { X 1 ( p ) + [ X 2 , X 3 ]( p ) | X 1 ( p ) , X 2 ( p ) ∈ H i ( p ) , X 3 ( p ) ∈ H ( p ) } We obtain a sequence H = H 1 ⊆ H 2 ⊆ · · · ⊆ H s = TM We say that ( M , H , g ) is regular if the dimension of H i ( p ) does not depend on p , for every i = 1, . . . , s . Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 4 / 14
Carnot-Carath´ eodory distance A smooth curve γ : [ 0, 1 ] → M is horizontal if γ ′ ( t ) ∈ H ( γ ( t )) for every t ∈ [ 0, 1 ] . We define � 1 0 g γ ( t ) ( γ ′ ( t ) , γ ′ ( t )) 1/2 dt l ( γ ) = the length of γ . For every p , q ∈ M d ( p , q ) = inf { l ( γ ) | γ ( 0 ) = p , γ ( 1 ) = q , γ horizontal } is a distance on M . Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 5 / 14
Left invariant subRiemannian manifolds If M = G , we have L p q = pq , and ( L p ) ∗ q : T q G → T pq G . Let g = Lie G . Fix V ⊆ g such that g = span { [ e 1 , [ e 2 , . . . , [ e i − 1 , e i ]]] | e i ∈ V } Let �· , ·� be a scalar product on V . Then H ( p ) : = ( L p ) ∗ e V g p ( v 1 , v 2 ) : = � ( L p − 1 ) ∗ p v 1 , ( L p − 1 ) ∗ p v 2 � define a left-invariant subRiemannian structure ( G , H , g ) . Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 6 / 14
Example g = span { e 1 , e 2 , e 3 } such that [ e 1 , e 2 ] = e 3 . exp g = Heisenberg group. subRiemannian structures: V 1 = span { e 1 , e 2 } , V 2 = span { e 3 } or V = span { e 1 , e 2 , e 3 } g = V 1 + · · · + V s s.t. [ V 1 , V i ] = V i + 1 , i = 1, . . . , s − 1, V s ⊂ z ( g ) . g is a nilpotent and stratified Lie algebra and exp g = G with C-C distance d is called a Carnot group. Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 7 / 14
subRiemannian isometries A mapping f : U → V with U , V ⊆ M open subsets is an isometry if d ( f ( p ) , f ( q )) = d ( p , q ) for every p , q ∈ U . Main questions: Smoothness of f . Characterize isometries. Example If ( M , d ) = ( R n , d E ) , then each isometry on U ⊆ R n is the restriction to U of { translation × O ( n ) } Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 8 / 14
State of the art Let G be a Carnot group. Theorem (Strichartz 1986, J. Differential Geom.) Suppose Lie G = g satisfies the strong bracket generating property. Let f : U ⊆ G → G be a smooth isometry from the open and connected subset U into G . If f ( e ) = e, then f = α | U , where α ∈ Aut G . We say that g satisfies the s.b.g.p. if ∀ X ∈ V 1 one has [ X , V 1 ] + V 1 = g ( ⇒ g is M´ etivi´ er). Theorem (Hamenst¨ adt 1990, J. Differential Geom.-Kishimoto 2003, J. Math. Kyoto Univ.) If f : G → G is an isometry such that f ( e ) = e, then f ∈ Aut G . Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 9 / 14
Main results Theorem (Le Donne, Ottazzi, 2013) Let f : U ⊆ G → G be an isometry, with U open. If f ( e ) = e, then f = α | U , where α ∈ Aut G . Theorem (Le Donne, Ottazzi, 2013) Let ( G / H , H , g ) be a homogeneous subRiemannian manifold such that g is G-invariant. Let f : G / H → G / H be an isometry. Then f is smooth. Moreover, f is uniquely defined by f ( o ) and df ( o ) | H ( o ) . Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 10 / 14
Remarks The statements remain true in the subFinsler setting. Namely, with � · � instead of �· , ·� . If G / H = G , it is not true in general that isometries are “affine” if G is not nilpotent and stratified. For example, in S 3 with the Euclidean distance the inversion map p �→ p − 1 is an isometry, but not a group isomorphism. Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 11 / 14
Proofs Regularity of isometries: Capogna, Cowling, 2006, Duke Math. J. ⇒ (local) isometries on Carnot groups are smooth. Montgomery Zipping theory on locally compact groups ⇒ global isometries on homogeneous manifolds are smooth. Studying the action of an isometry on Killing vector fields: Global vector fields. Unique extension of local vector fields in the Carnot group case (Tanaka prolongation theory). Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 12 / 14
Remark X locally compact, connected metric space, with finite dimension; 1 Iso ( X , d ) acts transitively on X . 2 ⇒ X is a differential manifold and Iso ( X , d ) is a Lie group. (Montgomery-Zippin, 1952) d is an intrinsic distance. 3 ⇒ X is a homogeneous subFinsler manifold (Berestovski˘ ı, 1989). ∃ λ > 1 s.t. ( X , d ) and ( X , λ d ) are isometric. 4 ⇒ X is a nilpotent and stratified Lie group (Mitchell 1985 - Margulis, Mostow 1995). Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 13 / 14
Open problems Regularity of isometries in general subRiemannian manifolds (Capogna and Le Donne solved this). Extend our results to the general setting of subRiemannian manifolds (with Capogna and Le Donne). Explicit characterization of isometries, e.g. on 3-dimensional Lie groups (with Le Donne). Alessandro Ottazzi (CIRM, Trento) Isometries of subRiemannian manifolds 14 / 14
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