Analysis of m-Modified Conformal Vector Fields on the Tangent Bundle With Synectic Lift Metrics

In this study, we introduce the notion of an m-modified conformal vector field, which represents a natural generalization of the classical conformal vector field concept. This extension allows for a broader class of vector fields that conform to modified transformation rules under conformal changes in geometry. Specifically, we explore the necessary and sufficient conditions under which the vector fields VV,VC,VC+γF, and uV (also known as the Liouville vector field) qualify as m-modified conformal vector fields within the framework of the synectic lift metric on the tangent bundle. The investigation into these conditions requires careful analysis using geometric tools and algebraic methods. In our calculations, we employ an adapted frame, which significantly streamlines the computations and provides multiple advantages for the geometric interpretations. Through this approach, we are able to derive deeper insights into the structure of these vector fields and their role within the broader context of differential geometry.