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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

**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:__

smooth or analytic manifold

distribution on (subbundle of , smooth or analytic)

inner product on , smooth or analytic

A submanifold is *horizontal* if

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

Whatever the , always horizontal *curves*: integral curves of vector fields in .

Maximal horizontal submanifolds? 2nd. half of talk.

Horizontal curves have

Carnot-Carathéodory "distance" on :

Main problems since :

Regularity of

Recovering from (Gromov)

Admissible domains for Bdy. Value Problems for

__Subelliptic Operators__

Defined by regularity condition: :

True for elliptic , where , wherefrom

(def. of hypoelliptic).

But false for ! (take arbitrary).

Important special case: sublaplacians

vector fields on a manifold

Classical: if is a basis of , then elliptic. This implies Existence, Uniqueness and Regularity for

Relation with previous situation:

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

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

Ultimate reason: is spanned by

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

**Classes of distributions**

tangent sheaf of (germs of vector fields)

subsheaf of fields .

Filtration of generated by :

__ V Involutive__: ().

__ V Outvolutive__, bracket-generating, fat:

Recall *Involutive* case:

Frobenius: *Involutive* *completely integrable:*

*with the leaves* *maximal.*

Involutive subriemannian geometry? Not very interesting: each is riemannian. distance between leaves is . is far from regular. Still, Lie algebras of vector fields are fundamental in Control. If is such, then not a distribution: dim may jump. It is a "Distribution with singularities", but involutive, so one can ask

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

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:

so span everywhere.

From now on, will be outvolutive.

THEOREMS. Assume can be connected with smooth arcs.

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

for any subriemannian structure on .

Regularity of is critical. For example, unless , *the function* *is not continuously differentiable in any punctured neighborhood of* *!*

Instead,

__Agrachev__:

*analytic** * *subanalytic*

i.e., -balls are subanalytic.

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

with real analytic.

Lojasiewicz, Sussmann, ...

As to subellipticity,

__Hormander__:

*If* *is a local basis of* *, then* *is hypoelliptic*

**Horizontal Submanifolds**

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**

satisfying

Canonical distribution on :

Origin: Gromov's Theorems on growth of discrete groups

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

is a sheaf of Carnot algebras.

How many? Even step 2

no classification is possible for . (Bernstein-Gelfand-Ponomarev-Gabriel-Coxeter-Dynkin Diagram has 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 , weakly convex gauge ...

Definition: with inner products such that

defined by

satisfies

Equivalently: defines unitary representation of Cliff() on .

Parametrized by 2 or 3 natural numbers

spinor spaces.

Analogy with symplectic.

History: fundamental solution for :

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

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* *and any point* *, there* *exists a unique horizontal subgroup* *such that*

*Any horizontal subgroup is abelian*.

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

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 dimensional Heisenberg, the distribution is 2n-dimensional and the maximal horizontal submanifolds are n-dimensional. Distinguished class: with totally isotropic in the usual sense.

In general, maximal possible dimension is : "Lagrangian" subspaces. Not always achieved. 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

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

Sometimes is a group. For example, if mod and

then

Always finite of -orbits, of the form , with reductive

NEXT: Maximal, but dim?

Examples! In Quaternionic 7-dimensional or Octonionic 15-dimensional Heisenberg there are no horizontal submanifolds of dimension . 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*