The configuration space for bicycling has a natural
rank 2 distribution as does the space of k-jets of a single function of a single
real variable. These distributions have natural inner products giving them the
structure of homogeneous subRiemannian manifolds. Their associated geodesic flows are integrable. The over-riding
question we pose is ``among all their geodesics which are metric lines?'' (A metric line is a bi-infinite globally minimizing geodesics.) Both geometries admit a metric submersion onto the Euclidean plane so `horizontal lifts' of Euclidean lines are always metric lines. The question becomes ``What else besides these lifted lines are metric lines?'' We completely answer the question for bicycling, the answer coming in the
form of one of Euler's elastica. In the jet space we have partial answers in terms of heteroclinic and homoclinic
orbits for associated one degree of freedom Hamiltonian systems. These systems are of standard kinetic plus potential type with potentials being squares of degree k polynomials. The question posed extends to the countably infinite family of Carnot groups on two generators (and beyond) so leaves the audience with a countable family of potential thesis and research problems, perhaps most of which are intractable. But maybe not.
The bicycling is joint work with Ardentov, Bor, LeDonne and Sachkov. See arxiv: 2010.04201
The jet space work in joint with Doddoli. See arXiv:2109.13835
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487