On affine motions with one-dimensional orbits in common spaces of paths

Abstract

The concept of a common path space was introduced by J. Duqlas. M. S. Knebelman was the first to consider affine and projective move­ments in these spaces. The general path space is a generalization of the space of affine connectivity. In this paper, we study spaces of paths that admit groups of affine motions with one-dimensional orbits. For each representation in the form of algebra of vector fields of the abelian Lie algebra and the Lr algebra containing the abelian ideal Lr-1, a system of equations of infinitesimal affine motions is compiled. The vector fields of each of these representations are operators of a group of transformations with one-dimensional orbits. Integrating this system, general spaces of paths are defined that admit a group of affine motions with one-dimensional orbits, the operators of which are the vector fields of these representations. The maximum order of these groups is set. It is shown that the spaces of paths admitting a group of affine motions with one-dimensional orbits of maximum order are projectively flat. The conditions that are necessary and sufficient for the space of paths to admit a group of affine motions with one-dimensional orbits of maximum order are given.

Keywords

• tangent bundle
• general path space
• a projectively flat space
• lie derivative
• infinitesimal affine transformation