Flattened infinitesimal transformations of the tangent bundle with horizontal lift connection, generated by infinitesimal holomorphically projective transformations

Background. The study of infinitesimal transformation lifts goes back to works by K. Yano and S. Ishihara. In particular, they reveal conditions at which the complete lift of infinitesimal projective transformation is the infinitesimal projective transformation with respect to the complete lift connection. Generally the geometrical nature of the complete lift of infinitesimal transformation has not been established by them. In S. G. Lejko's works, it has been presented within the limits of the flattening maps theory. The purpose of the given work is to study flattening properties of the complete lift of infinitesimal holomorphically projective transformation. The tangent bundle is considered as an affinely connected space with the horizontal lift connection. Materials and methods. The work uses methods of tensor algebra and analysis, the toolkit of the theory of lifts from the basis manifold into the tangent bundle. Added by the vector's field curve, the concept of flattening is introduced. From this point of view the flattening infinitesimal transformations ($r$-geodetic infinitesimal transformations according to S. G. Lejko's terminology) are defined and, also, their equations are obtained in the invariant form. The horizontal lift of the affine connection represents the affine connection on the tangent bundle with the nontrivial torsion. Therefore the new type of lifting from the basis manifold into the tangent bundle is introduced in order to study the flattening properties of the complete lift of infinitesimal holomorphically projective transformation. The received properties of the E-lift show the role it plays in covariant differentiation concerning of the horizontal lift connection. Results. The flattened infinitesimal transformations are considered from the point of view of the introduced concept of flattening added by the vector field's curve. The authors have obtained the toolkit to investigate the flattening properties of the complete lift of infinitesimal transformation with respect to the horizontal lift connection. Its use has allowed to reveal the flattening properties of the complete lift of infinitesimal holomorphically projective transformation with respect to the horizontal lift connection. Conclusions. The complete lift of the infinitesimal holomorphically projective transformation with respect to the horizontal lift connection is: (1) $1$-g.i.t. if and only if the basis infinitesimal transformation is affine; (2) $2$-g.i.t. if and only if the covector field that defines the infinitesimal holomorphically projective transformation is covariant constantly; (3) $r$-g.i.t. ($r=3,4,5$) if and only if minors of the $r$-order of the coefficient matrix, located in the first $r$-rows and six columns, equal to zero; (4) generally $6$-g.i.t.; (5) absolutely canonical $r$-g.i.t. ($r=3,4,5$) if $S_{2,...,r}(\nabla ^{r-1} \beta)=0$.