We consider the family of λ connections∇(λ) on a statistical manifold Mequipped with a pair of conjugate connections ∇ = ∇(1) and ∇∗ = ∇(−1), where the λ connection is defined as ∇(λ) = 1+λ 2 ∇ + 1−λ 2 ∇∗. This paper develops expressions for the vertical and horizontal distributions on the tangent bundle of the statistical manifold (M, g,∇(λ)) and introduces the concept of λ-adapted frames. We also derive the Levi–Civita connection CGb∇ (λ) of the tangent bundle TM, which is equipped with the Cheeger Gromoll-type metric CGg. We study the statistical structure (CGg, CG∇ (λ) ) on the tangent bundle TM, which is naturally induced from the statistical manifold (M, g,∇(λ)) . By introducing a para-holomorphic structure on the statistical manifold (M, g,∇(λ)) , we construct a para- Hermitian structure on the tangent bundle TM and examine its integrability. Finally, we present the conditions under which these bundles admit a para-holomorphic structure.
