Geometry of tangent bundles equipped with the deformed complete lift metric

Abstract

The geometric structure of the tangent bundle $(TM,g^{{\textsc{d}}})$ equipped with the deformed complete lift metric $g^{{\textsc{d}}}$ is investigated. Necessary and sufficient conditions are established for the vertical and horizontal lifts of vector fields to be conformal or Killing vector fields with respect to $g^{{\textsc {d}}}$. The conditions under which the tangent bundle, endowed with the horizontal (resp. complete) lift connection, admits a Codazzi or statistical structure are also established. The study also includes the analysis of infinitesimal affine transformations and geodesics associated with $g^{{\textsc {d}}}$. Furthermore, explicit examples are given on the Euclidean space to illustrate and validate the obtained characterization of geodesics on the tangent bundle endowed with the deformed complete lift metric.