\(P\)-geodesic transformations of the tangent bundles, induced by geodesic transformations of base manifolds
Infinitesimal geodesic transformations of the base manifold with a torsion-free affine connection does not include a similar transformation of its first and second order tangent bundles provided with complete lift affine connections. In the present article both the induced transformations and the groups of these are investigated from the point of view of \(p\)-geodesic mappings theory. An infinitesimal local transformation of an affinely connected manifold \(A_ n\) is called an infinitesimal \(p\)-geodesic transformation (i. \(p\)- g.t.) if it imports a flattening of order \(q\) (where \(q \leq p\)) to every geodesic curve \(\gamma: (t_ 0,t_ 1) \to A_ n\) at every point \(\gamma(t)\). The number \(q\) can depend on both the choice of a geodesic curve \(\gamma\) and the point on this curve. The number \(p\) is maximal among all possible numbers \(q\). \(P\)-geodesic diffeomorphism \(\pi\): \(A_ n \to \overline{A}_ n\) of two affine connected manifolds is defined by analogy. Let \(G_ r\) be a local \(r\)-dimensional Lie group of non-trivial (i.e., non-affine) geodesic transformations of the manifold \(A_ n\) with the torsion-free affine connections, which correspond to the operators \(X_ s\) and covectors \(\omega_ s\), \(s = 1,\ldots,r\). On the second order tangent bundle \(T_ 2(A_ n)\) with the complete lift connection \(\nabla^{ll}\) the following assertions then take a place: 1) the operators \(X^{ll}_ s\) generate an Abelian group \(G^ 0_ r\) whose generators are canonic i. 2-g.t. \(\tau_ 1(2)\); 2) the operators \(X^{ll}_ s\) generate a non-Abelian \(G^{ll}_ r\) whose generators are non-canonic i. 3-g.t. \(\tau_ 1(3)\) (some of them can be 2-geodesic of the basic type \(\tau_ 0(2)\), in the case of absolute parallelism of the corresponding covector \(\omega_ 2\)); 3) the group \(G^{ll}_ r\) is an \(r\)-dimensional Lie group of finite non- canonic 3-geodesic transformations of the first linear type (some transformations of this group can be 2-geodesic of the basic type, if only \(\nabla\omega_ s = 0\)). This group \(G^{ll}_ r\) is induced by the base group \(G_ r\). Similar results can be obtained for the tangent bundles of arbitrary order. Remark: In theorem 5, 1) must be \(X^ 0\).
scientific article; zbMATH DE number 484323
scientific article