IFHP Transformations on the Tangent Bundle with the Deformed Complete Lift Metric

Let ( M n , g ) be a Riemannian manifold and TM n the total space of its tangent bundle. In this paper, we determine the infinitesimal fiber-preserving holomorphically projective (IFHP) transformations on TM n with respect to the Levi-Civita connection of the deformed complete lift metric ˜ G f = g C + ( fg ) V , where f is a nonzero differentiable function on M n and g C and g V are the complete lift and the vertical lift of g on TM n , respectively. Morevore, we prove that every IFHP transformation on ( TM n , ˜ G f ) is reduced to an affine and induces an infinitesimal affine transformation on ( M n , g ) .