Geometry of the Second-Order Tangent Bundles ofRiemannian Manifolds

Let $(M,g)$ be an $n$-dimensional Riemannian manifold and $T^{2}M$be its second-order tangent bundle equipped with a lift metric$wt{g}$. In this paper, first, the authors construct someRiemannian almost product structures on $(T^{2}M,wt{g})$ andpresent some results concerning thesestructures. Then, they investigate the curvature properties of $(T^{2}M,wt{%g}).$ Finally, they study the properties of two metric connections withnonvanishing torsion on $(T^{2}M,wt{g})$: The $H$-lift of theLevi-Civita connection of $g$ to $T^{2}M,$ and the product conjugateconnection defined by the Levi-Civita connection of $wt{g}$ and analmost product structure.