Sub-Riemannian Geometry and Interactions – SRGI

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 these spaces.
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 distribution.
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.