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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006


Carnot manifolds

Aroldo Kaplan

Transparencies of the talk given at egeo2005 (La Falda, Junio de 2005), slightly edited




Subriemannian Geometry, Subelliptic Operators, Subanalytic Varieties, Subsymplectic Geometry.

Only eventually (not originally) related.

Motivations: PDE, Control Theory, non-holonomic systems, ...


Subriemannian Geometry:


M = smooth or analytic manifold

V = distribution on M (subbundle of T (M ) , smooth or analytic)

g = inner product on V , smooth or analytic


A submanifold N ⊂ M is horizontal if ∀p ∈ N

Tp(N ) ⊂ Vp.


∙ Must think that moving along non-horizontal directions is forbidden. Analogy with Parking Problem.

∙ Whatever the V , ∃ always horizontal curves: integral curves of vector fields in V .

∙ Maximal horizontal submanifolds? 2nd. half of talk.

∙ Horizontal curves γ : [a,b] → M  have

 ∫ b ∘ ------------- Lenght (γ) = g(γ′(t),γ ′(t)) dt a

Carnot-Carathéodory "distance" on M :

d(p,q) = inf{Lenght (γ)} γ

γ horizontal, γ (a) = p , γ(b) = q ,

d (p, q) = ∞

if ∃ no such curve.


∙ Main problems since ~ 1985 :

Regularity of d(p,q)

Recovering V from d (Gromov)

Admissible domains for Bdy. Value Problems for


Subelliptic Operators

Defined by regularity condition: ∃ε > 0 :

||f||H ε ≤ C (||Df ||H0 + ||f ||H0)

True for elliptic D , where ε = 1 , wherefrom

Df = g, g smooth ⇒ f smooth

(def. of hypoelliptic).

But false for ∂- ∂xf(x,y) = 0 ! (take f(x,y ) = φ(y) arbitrary).


Important special case: sublaplacians

D = X21 + ...+ X2k

Xi vector fields on a manifold M


∙ Classical: if  n {Xi(p)}i=1 is a basis of Tp (M ) ∀p , then D elliptic. This implies Existence, Uniqueness and Regularity for

Df = g


∙ Relation with previous situation:

Vp ↔ span {Xi(p)}


∙ Kohn's sublaplacian in  3 R (bdy. value of the Bargmann laplacian of the 3-ball in  2 C ):

 ( ∂ ∂ )2 ( ∂ ∂ )2 D = ∂x- + y∂z- + ∂y-- x ∂z-

not elliptic, degenerates along z -axis. Still, hypoelliptic, even analytically so:

 r r Df = g ∈ C ⇒ f ∈ C (r = ∞, ω )

Ultimate reason:  3 R is spanned by

 ∂ ∂ ∂ ∂ ∂ X = ---+ y --, Y = ---- x ---, [X, Y] = - 2--- ∂x ∂z ∂y ∂z ∂z


Remark: Kohn's sublaplacian can be viewed as the boundary value of the laplacian of the 3-ball in  2 C , with its Bargmann metric.


  Classes of distributions


T = tangent sheaf of M (germs of vector fields)


V = subsheaf of fields Xq ∈ Vq .


Filtration of T generated by V :

W1 = V W2 = V + [V, V] W3 = V + [V, V] + [V, [V, V ]] .. .


∙ V Involutive:  Wj = V ∀j ([V, V ] ⊂ V ).


∙ V Outvolutive, bracket-generating, fat: ∃kWk = T


Recall Involutive case:


∙ Frobenius: Involutive ⇒ completely integrable:

 ⊎ M = N α, Vp = Tp(N α(p)) α

with the leaves N α maximal.


Involutive subriemannian geometry? Not very interesting: each N α is riemannian. dCC distance between leaves is ∞ . ∑ X2 i is far from regular. Still, Lie algebras of vector fields are fundamental in Control. If L is such, then p ↦→ L(p) not a distribution: dimL (p) may jump. It is a "Distribution with singularities", but involutive, so one can ask


∙ Is every point of M contained in a unique maximal integral submanifold of L ?


Answer: NO for smooth, YES for analytic.


(Hermann-Nagano Theorem). This is why real analyticity - and, eventually, subanalyticity - eventually come in [S].

  Outvolutive distributions


∙ Example:  3 M = R

 -∂- -∂- -∂- -∂- V = span {X = ∂x + y∂z , Y = ∂y - x∂z }

 ∂-- [X, Y ] = - 2 ∂z ∕∈ V

so X, Y, [X, Y ], span Tp(R3) everywhere.


From now on, V will be outvolutive.


THEOREMS. Assume M can be connected with smooth arcs.


∙ Chow (anti-Frobenius). Any two points can be joined by a smooth horizontal curve.


⇒ d(p,q) < ∞ for any subriemannian structure on (M, V ) .


∙ Regularity of d(p,q) is critical. For example, unless V ⁄= T (M ) , the function p ↦→ d(po,p) is not continuously differentiable in any punctured neighborhood of p o !




∙ Agrachev:


(M, V,g) analytic ⇒ d(p ,p) o subanalytic


i.e., d -balls are subanalytic.


Subanalytic sets: locally projections of semianalytic. Equivalently, locally of the form

{x : fi(x,y) = 0, gj(x,y) ≥ 0}

with fi,gj, real analytic.


Lojasiewicz, Sussmann, ...


As to subellipticity,


∙ Hormander:


If Xi is a local basis of V , then ∑ X2 j is hypoelliptic


Horizontal Submanifolds

N `→ M : Tp(N ) ⊂ Vp


∙ ∃ many horizontal curves. Higher dimension?


Integral objects of non-integrable things are likely interesting. Also occur spontaneously in minimal surfaces, Jets of Maps, Control, ... But are hard to find, no general pattern.



  Carnot Groups


Lie(G ) = g = v1 ⊕ v2 ⊕ ...⊕ vk


[v1,vj] = vj+1

Canonical distribution on G :

V = v1


∙ Origin: Gromov's Theorems on growth of discrete groups


∙ Not just examples: any outvolutive distribution filters T M . The associated graded

 W ⊕ Gr = Wj ∕Wj+1 j

is a sheaf of Carnot algebras.


∙ How many? Even step 2

g = v ⊕ z z = [v, v]

no classification is possible for dim z > 3 . (Bernstein-Gelfand-Ponomarev-Gabriel-Coxeter-Dynkin Diagram has dim z edges joining 2 vertices)


Models of models?

  Groups of Heisenberg type


But "as role models go, they are hard to emulate": only Carnot groups with abundant domains admissible for the Dirichlet problem and/or explicit fundamental solutions for ∑ X2i , weakly convex gauge ...


∙ Definition: g = v ⊕ z with inner products such that

Jz : v → v

defined by

 ′ ′ (Jzu,u )v = (z,[u,u ])z


J 2z = - 2|z|2I.

Equivalently: Jz defines unitary representation of Cliff(z ) on v .

Groups of H type ≈ Clif ford modules

Parametrized by 2 or 3 natural numbers

g = (nSm ) ⊕ Rm


 + - m g = (n+S m ⊕ n- S m) ⊕ R

Sm, S+m,Sm-, spinor spaces.


∙ Analogy with symplectic.


∙ History: fundamental solution for  ∑ 2 D = X j :

D Φ = δ f or Φ (exp(v + z)) = -------C------- (|v|4 + 16|z|2)N

Since then keep yielding interesting riemannian examples (K., Willmore-Damek-Ricci, Selberg-Lauret, Gordon, Szabo, ... ).


∙ Aut(G ) is largest. As to subriemannian:


"Manifolds of Heisenberg type are to subriemannian Geometry, as Euclidean spaces, or symmetric spaces, are to riemannian geometry"


∙ The search for maximal horizontal submanifolds in groups of Heisenberg type is joint work with Levstein, Saal, Tiraboschi.


∙ In any Carnot group, for any horizontal submanifold N and any point p ∈ N , there exists a unique horizontal subgroup GN,p such that

Tp(N ) = (Lp)*(Te(GN,p)).

Any horizontal subgroup is abelian.


Hope to find all the latter. The following points to other reasons

M aximal isotropic subspaces of (v,[ , ]) U


M aximal horizontal subgroups of G exp(U )


M aximal abelian subgroups of G exp(U + Z )


∙ Classical examples to keep in mind:

On the 3-dimensional Heisenberg group, the distribution is 2-dimensional and the maximal horizontal submanifolds are 1-dimensional. But too many. Subgroups then.

On the 2n + 1 dimensional Heisenberg, the distribution is 2n-dimensional and the maximal horizontal submanifolds are n-dimensional. Distinguished class: exp (U) with U ⊂ v totally isotropic in the usual sense.


∙ In general, maximal possible dimension is dim v ∕2 : "Lagrangian" subspaces. Not always achieved. Lag (G) is a variety. Have a description (to be presented by Levstein in Colonia).


∙ Relation with Schroedinger Representations (?), Deligne's "Reality and the Heisenberg group".




∙ Sometimes Lag (G) = ∅  


∙ Sometimes any two Lagragians are conjugate by an automorphism of G , sometimes not.


∙ Sometimes Lag (G) is a group. For example, if m ≡ 7 mod 8 and

 + n - n m Lie(G ) = (Sm ) ⊕ (S m) ⊕ R


Lag(G ) = O (n)


∙ Always Lag (G ) = finite ⋃ of Aut (G) -orbits, of the form  ′ K ∕K , with  ′ K, K , reductive


∙ NEXT: Maximal, but dim< n ∕2 ?


Examples! In G = Quaternionic 7-dimensional or Octonionic 15-dimensional Heisenberg there are no horizontal submanifolds of dimension > 1 . The distribution has dimension 4 and 8, respectively.


A good starting point for geometers is R. Montgomery's review of Gromov's book. The following references were chosen specifically for the talk, but they and their bibliography are representative

R. Montgomery A tour of subriemannian geometries, their geodesics and applications, A.M.S. Mathematical Surveys and Monographs, 2002        [ Links ]

Gromov Carnot-Carathéodory spaces seen from within, in Subriemannian Geometry, Progr. Math., 144, Bikhauser (1996). Review by R. Montgomery in MathSciNet 2000f:53034        [ Links ]

Capogna - Garofalo - Nhieu, Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type, American Journal of Mathematics 124, 2 (2002)        [ Links ]

Danielli - Garofalo - Nhieu, Notions of convexity in Carnot groups, Comm. in Analysis and Geom., 11-2 (2003)        [ Links ]

M. Christ, A remark on sums of squares of complex vector fields, math.CV/0503506        [ Links ]

Kaplan Fundamental solutions for a class of hypoelliptic operators associated with composition of quadratic forms, Trans. A.M.S. 258 (1980)        [ Links ]

Agrachev - Gauthier On the subanalyticity of CC distances, Annales de l'Institut Henri Poincaré; Analyse non-linéare 18, No. 3, (2001), Review by Sussmann in MathSciNet 2002h:93031        [ Links ]

Sussmann, Why real analyticity is important in Control Theory, Perspectives in control theory, Birkhuser (1990)        [ Links ]

Citti - Sarti A cortical based model of perceptual completion in the roto-translation space, Workshop on Second Order Subelliptic Equations and Applications, Cortona, June 2003        [ Links ]

Aroldo Kaplan
FaMAF-CIEM, Universidad Nacional de Córdoba,
Córdoba 5000, Argentina

Recibido: 30 de septiembre de 2005
Aceptado: 27 de septiembre de 2006