Project IV 2026-27

Sub-Riemannian Geometry and Applications

Martin Kerin

Credit: Photo by Frank Cone, Pexels.com.

Description This project will explore sub-Riemannian geometry, a generalisation of Riemannian geometry in which distances between points are measured using only curves tangent to a prescribed distribution of directions. In other words, movement within the space is restricted to certain directions. This constraint alters the geometry of the manifold, modifying classical Riemannian notions such as distance and geodesics. It also gives rise to intriguing phenomena, for example, the topological and Hausdorff dimensions may differ, as seen in simple cases like the Heisenberg group. Beyond their intrinsic mathematical interest, sub-Riemannian manifolds arise naturally in geometric mechanics, where they model systems such as robot motion or the dynamics of orbiting bodies. Project Research Area Pure Mathematics Mode of Operation and Evidence of Learning The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats. Prerequisites Essential: Analysis in Many Variables II Topology II Recommended: Differential Geometry III Corequisites Essential: None Recommended: Riemannian Geometry IV Some Resources Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, AMS, 2002. Andrei Agrachev, Davide Barilari and Ugo Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, vol. 181, Cambridge University Press, 2019. Enrico Le Donne, Lecture notes on sub-Riemannian geometry: Carnot-Carathéodory spaces from the Lie group viewpoint, Springer, 1979. John P. D’Angelo and Jeremy T. Tyson, An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries, Notices Amer. Math. Soc. 57 (2010), no. 2, 208–219. Additional Information If you would like more information about this project, please contact me at martin.p.kerin@durham.ac.uk