منیفلدهای دو-هم‌تافته‌ی آماری و زیرمنیفلدهای آن‌ها

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.



Moreover, an affine and torsion free connection ∇* is called a dual connection with respect to g, if

UgV,W=g∇UV,W+gV,∇*U W.

An almost contact manifold (M,φ,ξ,η) with Riemannian metric g is an almost contact-like manifold if it has another (1,1)-tensor field φ* satisfying

gφU,V=-gU,φ*V, gU, ξ= ηU.

Let (M,φ,ξ,η) be an almost contact-like manifold, then for all U,V∊τ(M) the following relations hold

gφU, φ*V=gU,V- ηUηV, φ*2U=-U+ηUξ.

Definition. An almost contact-like manifold (M,∇,φ,ξ,η,g) with statistical structure (∇,g) is a statistical almost contact-like manifold. Moreover, M,∇,φ,ξ,η,g is called a statistical cosymplectic manifold if

∇UφV=0.

M is an invariant submanifold of a statistical cosymplectic manifold M,∇,φ,ξ,η,g, if for all U∊τ(M) we have φU∊τM, φ*U∊τM.

Submanifolds of statistical cosymplectic manifolds

We show that the manifold M,∇,φ,ξ,η,g is a statistical cosymplectic manifold if and only if (M,∇*,φ*,ξ,η,g) is a statistical cosymplectic manifold. Moreover we prove the following theorems.

Theorem. Any invariant submanifold of a statistical cosymplectic manifold with tangent structure vector field ξ, is a statistical cosymplectic and minimal-like submanifold.

Theorem. Let M be a submanifold of statistical cosymplectic manifold (M,∇,φ,ξ,η,g) such that the structure vector field ξ is normal to M. Then for any vector field U∊τ(M) we have

A*ξU=0, ∇⊥Uξ=η∇Uξξ.

Theorem. Let M,∇,φ,ξ,η,g be a statistical cosymplectic manifold. If M is a submanifold of M and the structure vector field ξ is normal to M then

R⊥U,Vξ=0, ∀U,V∊τM.

Theorem. Let M be an invariant submanifold of statistical cosymplectic manifold M,∇,φ,ξ,η,g and ξ is normal to M. Then M is a statistical Keahler-like manifold.

Conclusion

We introduce statistical cosymplectic manifolds and investigate some properties of their tensors. We define invariant and anti-invariant submanifolds and study invariant submanifolds with normal and tangent structure vector fields. 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 show if the structure vector field is normal to the submanifold then that is a statistical Keahler-like manifold