The projective curvature of the tangent bundle with natural
Abstract
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 sufficient
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).
the total space $TM$ of the tangent bundle of a Riemannian
manifold $(M,g)$. We provide the necessary and sufficient
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).
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