Statistical cosymplectic manifolds and their submanifolds - Mohammad Bagher Kazemi; Shiva Salahvarzi (پژوهش‌های ریاضی, 2022)
Introduction Let p(x,ζ) be the set of parametric probability distribution with parameter ζ=ζ1,…,ζn∊Rn. This set is called a statistical model or manifold. The distance between two points is measured by the Fisher metric. In general, statistical manifolds are Riemannian manifolds of distributions endowed with the Fisher information metric. On the other hand, one of the most important structures on odd dimensional Riemannian manifolds is the almost contact structure. Recently, statistical manifolds equipped with almost contact structures are studied by many authors. In this paper, we introduce statistical almost contact-like and statistical cosymplectic manifolds on a Riemannian manifold. We recall the basic definitions and define statistical cosymplectic manifolds and their invariant submanifolds. We prove that an invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field is a cosymplectic and minimal-like submanifold. Also, we prove if the structure vector field be normal to the submanifold then the submanifold is a statistical Keahler-like manifold. Finally, we construct two examples to illustrate some results of the paper. Statistical almost contact-like manifolds Let M,g be a Riemannian manifold with the Levi-Civita connection ∇. M,g is called a statistical manifold if there exists an affine and torsion free connection ∇ such that for all U,V,W∊τM ∇UgV,W=∇VgU,W.