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

TL;DR

This paper explores the geometric properties of lift metrics and connections on the tangent bundle of a Riemannian manifold, examining their implications for statistical structures and special curvature conditions.

Contribution

It introduces new results on lift metrics, connections, and their impact on statistical and Codazzi structures, as well as conditions for 1-Stein and Osserman structures on tangent bundles.

Findings

Analysis of lift metrics and connections on tangent bundles

Implications for statistical and Codazzi structures

Conditions for 1-Stein and Osserman structures with complete lift connection

Abstract

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.