Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric

We consider the family of λ connections ∇ ( λ ) on a statistical manifold M equipped 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 ∇ ^ C G ( λ ) of the tangent bundle T M , which is equipped with the Cheeger Gromoll-type metric g C G . We study the statistical structure ( g C G , ∇ C G ( λ ) ) on the tangent bundle T M , 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 T M and examine its integrability. Finally, we present the conditions under which these bundles admit a para-holomorphic structure.