Geometry of the second-order tangent bundles of Riemannian manifolds

Abstract

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

Keywords

Almost product structure / Killing vector field / Metric connection / Riemannian metric / Second-order tangent bundle

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