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We study lift metrics and lift connections on the tangent bundle $TM$ of aRiemannian manifold $(M,g)$. We also investigate the statistical and Codazzicouples of $TM$ and their consequences on the geometry of $M$. Finally, weprove a result on $1$-Stein and Osserman structures on $TM$, whenever $TM$ isequipped with the complete lift connection.

On the geometry of lift metrics and lift connections on the tangent bundle