Path geometries and almost Grassmann structures. (English) Zbl 1168.53010

Sabau, Sorin V. (ed.) et al., Finsler geometry, Sapporo 2005. In memory of Makoto Matsumoto. Proceedings of the 40th Finsler symposium on Finsler geometry, Sapporo, Japan, September 6–10, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-42-6/hbk). Advanced Studies in Pure Mathematics 48, 225-261 (2007).
The authors describe the association of an almost Grassmann geometry to each path geometry. They provide a clear description of the relationship between the path geometry invariants of Thomas and the Cartan connection used in more recent works of Tanaka, Fels, Grossman, Nurowski and others. Their work is carried out in an arbitrary local coordinate system.
A smooth family of paths on a manifold, one through each point in each direction, is called a path geometry. It is well known that each path geometry on a manifold \(M\) determines a foliation, and also a Cartan connection, on the sphere bundle \(SM\). The Cartan connection is modelled on \(ST \mathbb{RP}^n\), the sphere bundle of real projective space. This sphere bundle is a covering space of the Grassmannian of lines in projective space, i.e. the Grassmannian \(\text{Gr}_2\left(n+1\right)\) of 2-planes in \(\mathbb{R}^{n+1}\). Any Cartan connection modelled on \(ST \mathbb{RP}^n\) is by definition also modelled on \(\text{Gr}_2\left(n+1\right)\). Therefore any Cartan connection with such a model is refered to as an almost Grassmann structure.
The authors explain how to calculate the Cartan connection of any path geometry explicitly in coordinates. They express the components of the curvature in terms of known invariants of path geometries. They determine which almost Grassmann structures arise from path geometries. They provide a test to determine whether a path geometry is the geodesic geometry of a Finsler metric (or a more general pseudo-Finsler function).
For the entire collection see [Zbl 1130.53005].
Reviewer: Ben McKay (Cork)


53B10 Projective connections
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry