We study lift metrics and lift connections on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$. We also investigate the statistical and Codazzi couples of $TM$ and their consequences on the geometry of $M$. Finally, we prove a result on $1$-Stein and Osserman structures on $TM$, whenever $TM$ is equipped with the complete lift connection.
PEYGHAN, ESMAEIL; SEIFIPOUR, DAVOOD; and BLAGA, ADARA
"On the geometry of lift metrics and lift connections on the tangent bundle,"
Turkish Journal of Mathematics: Vol. 46:
6, Article 20.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss6/20