منیفلدهای دو - هم‌ تافته‌ی آماری و زیرمنیفلدهای آنها

در این مقاله منیفلدهای تقریبا تماسی خاص و دوهم ‌تافته‌ آماری را تعریف کرده و برخی از خواص تانسورهای آن‌‌ها را بررسی می‌نماییم. ضمن معرفی زیرمنیفلدهای پایا و پاد پایا، به مطالعه‌ی زیرمنیفلدهای پایا با میدان برداری ساختاری مماس و نرمال می‌پردازیم. به ویژه ثابت می‌کنیم هر زیرمنیفلد پایای یک منیفلد دو هم‌ تافته‌ آماری‌ با میدان برداری ساختاری مماس، دوهم تافته آماری و شبه مینیمال است و اگر میدان برداری ساختاری نرمال ‌باشد، زیرمنیفلد شبه کیلری آماری است. به علاوه با ساختن مثالی غیر بدیهی، درستی موارد فوق را در آن نشان می‌دهیم.

statistical cosymplectic manifolds and their submanifolds

introductionlet 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 contactlike 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 minimallike submanifold. also, we prove if the structure vector field be normal to the submanifold then the submanifold is a statistical keahlerlike manifold. finally, we construct two examples to illustrate some results of the paper.statistical almost contactlike manifoldslet m,g be a riemannian manifold with the levicivita 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 contactlike manifold if it has another (1,1)tensor field φ* satisfyinggφu,v=gu,φ*v,              gu, ξ= ηu.let (m,φ,ξ,η) be an almost contactlike manifold, then for all u,v∊τ(m) the following relations holdgφu, φ*v=gu,v ηuηv,  φ*2u=u+ηuξ.definition. an almost contactlike manifold (m,∇,φ,ξ,η,g) with statistical structure (∇,g) is a statistical almost contactlike 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  minimallike 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 havea*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 thenr⊥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 keahlerlike manifold.conclusionwe introduce statistical cosymplectic manifolds and investigate some properties of their tensors. we define invariant and antiinvariant 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 minimallike submanifold. also we show if the structure vector field is normal to the submanifold then that is a statistical keahlerlike manifold