versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
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, ...
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 :
Recovering from (Gromov)
Admissible domains for Bdy. Value Problems for
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].
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 !
i.e., -balls are subanalytic.
Subanalytic sets: locally projections of semianalytic. Equivalently, locally of the form
with real analytic.
Lojasiewicz, Sussmann, ...
As to subellipticity,
If is a local basis of , then is hypoelliptic
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.
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
Equivalently: defines unitary representation of Cliff() on .
Parametrized by 2 or 3 natural numbers
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".
Sometimes any two Lagragians are conjugate by an automorphism of , sometimes not.
Sometimes is a group. For example, if mod and
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.
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 ]
FaMAF-CIEM, Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
Recibido: 30 de septiembre de 2005
Aceptado: 27 de septiembre de 2006