A Short Note on a Mus-Cheeger-Gromoll Type Metric

In this paper, we first show that the complete lift $U^{c}$ to $TM$ of a vector field $U$ on $M$ is an infinitesimal fiber-preserving conformal transformation if and only if $U$ is an infinitesimal homothetic transformation of $(M,g)$. Here, $(M, g)$ is a Riemannian manifold and $TM$ is its tangent bundle with a Mus-Cheeger-Gromoll type metric $\\tilde{g}$. Secondly, we search for some conditions under which $\\left(\\overset{h}{\\nabla},\\tilde{g}\\right)$ is a Codazzi pair on $TM$ when $(\\nabla, g)$ is a Codazzi pair on $M$ where $\\overset{h}{\\nabla}$ is the horizontal lift of a linear connection $\\nabla$ on $M$. We finally discuss the need for further research.