2018
Том 70
№ 9

Zubrilin K. M.

Articles: 2
Article (Russian)

On Preservation of the Order of Flattening by an Induced Diffeomorphism

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1482–1497

We consider the structure of a smooth curve from the viewpoint of the concept of flattening and establish conditions under which an r-geodesic curve of the base manifold is the projection of the r-geodesic curve in a tangent bundle of the second order. The necessary and sufficient condition under which a 2-geodesic diffeomorphism of affine-connected spaces induces a 2-geodesic diffeomorphism of tangent bundles of the second order is established.

Article (Russian)

p-Geodesic transformations and their groups in second-order tangent bundles induced by concircular transformations of bases

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 346-364

We investigate the flattening properties of the Lie group $G_r^{II}$ of transformations of a second-order tangent bundle $T^2(M)$ equipped with the lift $∇^{II}$ of an affine connection $∇$ and the lift $g^{II}$ of a metric $g$ on the base of $M$ induced by the Lie group $G_r$ of concircular transformations of the base of $M$. The obtained results reveal certain geometric features of the induced group $G_r^{II}$ within the framework of the theory of $p$-geodesic mappings.