The geometry of the tangent and sphere bundles over statistical manifolds

In the paper a Riemannian structure on the tangent bundle is defined by using a statistical structure $(g,\mathrm{\nabla })$ on the base manifold. Expressions for various curvatures of the structure are derived. Some rigidity results of the structure are proved. The main goal of the paper is to initiate the study of sphere bundles over statistical manifolds. Basic formulas for the geometry are established. Sphere bundles with small radii over compact manifolds are studied.