Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles

Abstract

In this paper, we consider a second-order tangent bundle equipped with
 Sasaki metric over a Riemannian manifold. All forms of curvature tensor
 fields are computed. We obtained the relation between the scalar curvature
 of the base manifold and the scalar curvature of the second-order tangent
 bundle and presented some geometric results concerning with kinds of
 curvature tensor fields. Also, we search the weakly symmetry property of the
 second-order tangent bundle. Finally, we end our paper with statistical
 structures on the second-order tangent bundle.