Chains of path geometries on surfaces: theory and examples

具有常挠率的锥结构与路径几何

Cone structures and path geometries with constant torsion

本文研究了具有非平凡最大秩挠率的三维路径几何，引入常挠率概念，证明其与齐次直纹曲面上的锥结构一一对应，并通过新可积系统的解进行描述。

Contact Path Geometries

Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's subject to a single maximally non-integrable constraint. A subclass of contact path geometries is distinguished by the vanishing of an invariant contact torsion. For this subclass the equivalence problem i…

路径几何的Weyl结构

Weyl structures for path geometries

本文提出了一种基于Weyl结构的路径几何研究方法，通过定义一类特殊连接和Schouten张量，为路径几何的tractor算术提供了基础，并探讨了其在不变算子构造中的应用。

On the Geometry of Chains

The chains studied in this paper generalize Chern-Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures. We develop a general method for extending Cartan geometries which generalizes the Cartan geometry i…

Metrisability of three-dimensional path geometries

Given a projective structure on a three-dimensional manifold, we find explicit obstructions to the local existence of a Levi-Civita connection in the projective class. These obstructions are given by projectively invariant tensors algebraically constructed from the projective Weyl curvature. We show, by examples, that their vanishing is necessary but not sufficient for local metrisability.

Local embeddability of real analytic path geometries

An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler theorem we show that locally every real analytic path geometry is induced by an embedding into C^2 equipped with the splitting generated by the real and imaginary part of the standard holomorphic volume form. As a corollary we obtain the well-known fact that every 3-dimensional …