Affine transformations of the tangent bundle of a common path space

Abstract

In this paper, we study infinitesimal transformations of the tangent bundle of a common path space. The general path space is a genera­liza­tion space of the affine connectivity. By affine connectivity of the com­mon path space, we construct an affine connection on the tangent bundle. For the infinitesimal transformation of the tangent bundle, a system of in­va­riance equations for the constructed affine connectivity is compiled. This system is a system of second-order differential equations with res­pect to the components of the infinitesimal transformation. The main re­sults of the article are obtained by analyzing this system taking into account the properties of homogeneous functions. It is shown that the comp­lete lift of an infinitesimal transformation of base is an infinitesimal af­fine motion of a tangent bundle if and only if the infinitesimal transfor­ma­tion of base is an affine motion in the general path space. Necessary and sufficient conditions are found that the infinitesimal transformation of a tangent bundle generated by a vertical vector field leaves the affine connec­ti­vity of the tangent bundle invariant. Conditions are given that are neces­sa­ry and sufficient so that the infinitesimal transformation of a tangent bundle with affine connectivity that preserves layers is an affine motion.

Keywords

• tangent bundle
• general path space
• lie derivative
• infini­te­si­mal affine transformation