Analytic torsion of (2,3,5) distributions

The physical system of a ball rolling on a surface without slipping and twisting gives rise to an intriguing 5-dimensional geometry. All possible configurations of a ball touching a surface form a 5-dimensional manifold. The no slipping and twisting condition can be encoded using a so-called (2,3,5) distribution on this configuration space, which is also known as generic rank two distribution in dimension five. The history of this geometry can be traced back to a seminal paper by Élie Cartan from 1910. The past two decades have seen a surging interest in the geometry of (2,3,5) distributions which was predominantly focused on studying their local properties. In this research project we study global aspects of (2,3,5) distributions by investigating their spectral properties. A (2,3,5) distribution gives rise to a sequence of natural differential operators called the Rumin complex. The spectrum of the Rumin complex consists of a discrete set of frequencies encoding global aspects of the geometry in a subtle manner. We analyze two spectral invariants extracted from these frequencies: the analytic torsion and the eta invariant. Analogues of these spectral invariants in Riemannian geometry have been studied thoroughly. Celebrated results relate them to topological and geometrical properties of the Riemannian metric. We will use the sub-Riemannian limit technique to infer results for (2,3,5) distributions from the Riemannian case. This method has proven itself well in the context of other geometries. A main goal of this project is the adaptation of the sub-Riemmannian limit technique to (2,3,5) distributions. The basic idea here is to rescale the Riemannian metric in such a way that the geometry of the (2,3,5) distribution is reflected in the limiting behavior. A fundamental question motivating our research can be phrased as follows: Which 5-dimensional manifolds support a (2,3,5) distribution? There are well-known topological obstructions to their existence. We would like to understand if geometry imposes further, more subtle obstructions.