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+λ2∇+1−λ2∇*. 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 ∇^CG(λ) of the tangent bundle TM, which is equipped with the Cheeger Gromoll-type metric gCG. We study the statistical structure (gCG,∇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.