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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

 

SUBMATH:

 

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 !

 

Instead,

 

∙ 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.

 

Models:

  Carnot Groups

 

Lie(G ) = g = v1 ⊕ v2 ⊕ ...⊕ vk

satisfying

[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

satisfies

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

or

 + - 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".

 

SOME CONCLUSIONS

 

∙ 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

then

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.

Bibliography

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
kaplan@mate.uncor.edu

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