Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M, g) be an n-dimensional Riemannian manifold and (TM)-M-2 be its second order tangent bundle equipped with a lift metric (g) over tilde. In this paper, first, the authors construct some Riemannian almost product structures on ((TM)-M-2, (g) over tilde) and present some results concerning these structures. Then, they investigate the curvature properties of ((TM)-M-2, (g) over tilde Finally, they study the properties of two metric connections with nonvanishing torsion on ((TM)-M-2, (g) over tilde): The H-lift of the Levi-Civita connection of g to (TM)-M-2, and the product conjugate connection defined by the Levi-Civita connection of (g) over tilde and an almost product structure.