Sub-Riemannian manifolds provide an
important mathematical model for many problems involving some
nonholonomic constraints. Since several years, there has been an
impressive revival of interest in sub-Riemannian geometry (in short, SR
geometry), together with many emerging interactions. SR geometry is of
course interesting in itself and raises a number of fascinating
mathematical challenges, such as the study of SR heat kernels,
hypoelliptic diffusions, volumes and singularities. But SR geometry has
also a wide potential of interactions. A classical (we could almost
say, historical) field of applications of SR geometry is robotics and
motion planning, in relationship with geometric control theory, but in
the recent years SR geometry has appeared relevant in new fields, with
striking issues, such as in optimal transport, in image reconstruction
and geometry of vision, or even more recently, in quantum physics and
in shape analysis.
The study of heat diffusion in Riemannian geometry has been strongly
developed, both from the probabilistic point of view (Brownian motions,
Wiener measures), and for the study of direct and inverse spectral
problems (Kac's famous ``can one hear the shape of a drum?'', Weyl
asymptotic formula). A similar study in the SR case (i.e., for SR
Laplacians) is still rather incomplete and important for applications.
Generalizing the definition of Laplace-Beltrami operator in Riemannian
geometry, SR Laplacians are defined as the divergence (for some
measure) of the horizontal gradient (for some SR metric).
In this perspective, a domain which is of interest to several
mathematical communities concerns the study of the small-time SR heat
kernel asymptotics, with many interesting issues that we want to
investigate: local Weyl law and quantum ergodicity within the SR
context, study of Weyl measures, propagation of singularities for SR
wave equations and impact of abnormal geodesics, inverse spectral
problems and identification of new spectral invariants.
This is the content of Task 1 of our project.
SR Laplacians depend on the choice of the measure, and in SR geometry
there are several intrinsic choices, such as Hausdorff or Popp volumes.
The regularity of those volumes, one with respect to each other, raises
surprisingly difficult questions, in particular at singularities of the
horizontal distribution of the SR structure. Such singularities cause
barrier phenomena for SR heat and Schrodinger flows. The (new) notion
of SR curvature seems to be related to spectral invariants that we want
to identify. The case of Carnot groups is of a particular interest. We
want also to investigate the concept of horizontal holonomy, which is
the SR version of the holonomy group associated with a connection.
This is the content of Task 2.
Task 3 gathers interactions of SR geometry with other fields.
The first consists of transportation problems involving nonholonomic
constraints. The problem of existence and uniqueness of an optimal
transport map on a general complete SR manifold is open. It is related
to curvature phenomena in SR manifolds and to isoperimetric problems in
The second is that SR geometry (and in particular, hypoelliptic
diffusion) provides a relevant framework to geometry of vision, as it
has been recently discovered, due to the fact that the architecture of
connections in the visual cortex seems to reflect a contact
Finally, even more recently SR geometry has emerged as well in shape
analysis, where pattern matching is achieved in the group of so-called
horizontal diffeomorphisms, thus modelling image analysis problems in
which motions are submitted to nonholonomic constraints.