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.
Reference
1. Douglas, J.: The general geometry of paths. Annalas of Math., 29, 143—168 (1928).
2. Yano, K.: The theory of Lie derivatives and its applications. Amsterdam (1957).
3. Yano, K., Isihara, S.: Tangent and Cotangent Bundles Differential Geometry. New York (1973).
4. Knebelman, M. S.: Collineations and motions in generalized spaces. Amer. J. Math., 51, 527—564 (1929).
5. Kobayashi, Sh., Nomizu, K.: Fundamentals of differential geometry, 1. Moskow (1981).
6. Nikitin, N. D.: On affine motions in general spaces of path. Izvestia vuzov. Math., 2, 21—25 (1996).
7. Nikitin, N. D.: On projective movements in common spaces of path. DGMF, 43, 100—107 (2012).
8. Nikitin, N. D., Nikitina, O. G.: Affine transformations of the tangent bundle of a common path space. DGMF, 54:2, 18—26 (2023).
9. Okubo, T.: On the order of the groups of affine collineations in the generalized spaces of paths. I. Tensor, 6, 141—158 (1956).