拉格朗日展开定理的两个基本应用

The present paper offers two likely neglected applications of the classical Lagrange expansion formula.One is a unified approach to some age-old derivative identities originally due to Pfaff and Cauchy.Another is two explicit matrix inversions which may serve as common generalizations of some known inverse series relations.     

Global properties of linear constraints in state space and motion planning

Study of nonholonomic motion planning needs further research into the global properties of linear constraints in state space. The global properties of constraints, which contain the holonomicity and the nonholonomicity by regions, the existence of the isolated integral manifolds and the singular points and so on, have essential influence on motion planning. By analysis of the point sets in total space, the complete sketch of the global properties of linear constraints in state space is obtained, which can directly be applied to motion planning. Project supported by the National Natural Science Foundation of China and Doctoral Programme Foundation Institution of High Education of China.     

Distributions of corank 1 and their characteristic vector fields

We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is invariantly assigned to a corank one distribution. It is defined on , if , or on a subset of called the Martinet hypersurface, if . Our second main result states that if two corank one distributions have the same characteristic line field and are close to each other, then they are equivalent via a diffeomorphism. This holds under a weak assumption on the singularities of the distributions. The second result implies that the abnormal curves of a distribution determine the equivalence class of the distribution, among distributions close to a given one.

     

Sur les systèmes de Pfaff totalement réguliers

Sans résumé

---

This article is the text of a talk given at the Symposium on differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.