IPHP Transformations on Tangent Bundle of a Riemannian Manifold with Respect to a Class of Lift Metrics

Abstract

Let \((M_n, g)\) be an \(n\)-dimensional Riemannian manifold and \(TM_n\) its tangent bundle. In this article, we study the infinitesimal paraholomorphically projective (IPHP) transformations on \(TM_n\) with respect to the Levi-Civita connection of the pseudo-Riemannian metric \(\tilde{g} = \alpha g^S + \beta g^C + \gamma g^V\), where \(\alpha, \beta\) and \(\gamma\) are real constants with \(\alpha(\alpha + \gamma) - \beta^2 \ne 0\) and \(g^S, g^C\) and \(g^V\) are diagonal lift, complete lift and vertical lift of \(g\), respectively. We determine this type of transformations and then prove that if \((TM_n, \tilde{g})\) has a non-affine infinitesimal paraholomorphically projective transformation, then \(M_n\) and \(TM_n\) are locally flat.