Our study is mainly devoted to a natural diagonal metric $G$ on the total space $TM$ of the tangent bundle of a Riemannian manifold $(M,g)$. We provide the necessary and sucient conditions under which $(TM,G)$ is a space form, or equivalently $(TM,G)$ is projectively Euclidean. Moreover, we classify the natural diagonal metrics $G$ for which $(TM,G)$ is horizontally projectively flat (resp. vertically projectively flat).