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منیفلدهای دو-همتافتهی آماری و زیرمنیفلدهای آنها
Statistical cosymplectic manifolds and their submanifolds
هندسه دیفرانسیل
Differential Geometry
علمی پژوهشی بنیادی
S
در این مقاله منیفلدهای تقریبا تماسی خاص و دو-همتافتهی 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- همتافتهی آماری با میدان برداری ساختاری مماس، دو-همتافته 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.<span dir="LTR"></span>
<span style="font-size:12pt"><span style="line-height:22.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Introduction</span></span></b></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span style="font-family:NimbusRomNo9L-Regu">Let </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><i><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>x</m:r><m:r>,</m:r><m:r>ζ</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span style="font-family:NimbusRomNo9L-Regu"> be the set of parametric probability distribution with parameter</span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r> </m:r></span></span></i><span dir="RTL" lang="AR-SA" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ζ</m:r></span></span><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ζ</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>1</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,…,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ζ</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sup></m:ssup></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>∊</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>R</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sup></m:ssup></m:omath><span style="font-size:9.0pt"><span style="font-family:NimbusRomNo9L-Regu">. 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. </span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span style="font-family:NimbusRomNo9L-Regu"> 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 </span></span><span style="font-size:9.0pt">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.</span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:17.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt">Statistical almost contact-like manifolds</span></b></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt">Let </span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>g</m:r></span></span></i></m:e></m:d></m:omath><span style="font-size:9.0pt"> be a Riemannian manifold with the Levi-Civita connection </span><m:omath><m:acc><m:accpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:accpr><m:e><span dir="RTL" lang="FA" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:omath><span style="font-size:9.0pt">. </span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>g</m:r></span></span></i></m:e></m:d></m:omath><span style="font-size:9.0pt"> is called a statistical manifold if there exists an affine and torsion free connection </span><m:omath><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:omath><span style="font-size:9.0pt"> such that for all </span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>U,V,W∊</m:r></span></span><span dir="RTL" lang="FA" style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>τ</m:r></span></span><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc></m:e></m:d></m:omath><span style="font-size:9.0pt"></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>∇</m:r></span></span></i></m:e></m:acc></m:e><m:sub><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e></m:d><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r><m:r>,</m:r><m:r>W</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>∇</m:r></span></span></i></m:e></m:acc></m:e><m:sub><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r></span></span></i></m:sub></m:ssub><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e></m:d><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>,</m:r><m:r>W</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></i></m:omath></m:omathpara><span lang="FA" style="font-size:9.0pt"><span style="font-family:"B Nazanin""></span></span></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Moreover, an </span></span><span style="font-size:9.0pt">affine and torsion free connection </span><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:e><m:sup><i><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup></m:omath><span style="font-size:9.0pt"> is called a dual connection with respect to </span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:omath><span style="font-size:9.0pt">, if </span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>Ug</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r><m:r>,</m:r><m:r>W</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>∇</m:r></span></span></i></m:e></m:acc></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r><m:r>,</m:r><m:r>W</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>+</m:r><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r><m:r>,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r> </m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>W</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></i></m:omath></m:omathpara><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">An almost contact manifold </span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r><i>(</i></m:r></span></span><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> with Riemannian metric </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is an almost contact-like manifold if it has another (1,1)-tensor field </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> satisfying</span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omathparapr><m:jc m:val="center"></m:jc></m:omathparapr><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>φ</m:r><m:r><i>U</i></m:r><m:r><i>,</i></m:r><m:r><i>V</i></m:r></span></span></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=-</m:r><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>, </m:r><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>, </m:r><m:r>ξ</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>= </m:r><m:r>η</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></i></m:omath></m:omathpara><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> be an almost contact-like manifold, then for all </span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>U,V∊</m:r></span></span><span dir="RTL" lang="FA" style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>τ</m:r></span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> the following relations hold</span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omath><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>φ</m:r><m:r><i>U</i></m:r><m:r><i>, </i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>g</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>,</m:r><m:r>V</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r><m:r> </m:r><m:r>η</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>η</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>, </m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*2</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>=-</m:r><m:r>U</m:r><m:r>+</m:r><m:r>η</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>.</m:r></span></span></m:omath></m:omathpara><span dir="LTR" style="font-size:9.0pt"></span></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Definition. </span></span></b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">An almost contact-like manifold </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> with statistical structure </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>g</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is a statistical almost contact-like manifold. Moreover, </span></span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i></m:e></m:d></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is called a statistical cosymplectic manifold if </span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omathparapr><m:jc m:val="center"></m:jc></m:omathparapr><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>V</m:r><m:r>=0.</m:r></span></span></i></m:omath></m:omathpara><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span new="" roman="" style="font-family:" times=""><m:omath><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span dir="LTR" style="font-size:9.0pt"> is an invariant submanifold of a sta</span><span dir="LTR" style="font-size:9.0pt">tistical cosymplectic manifold</span> <m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i></m:omath><span dir="LTR" style="font-size:9.0pt"> if for all </span><m:omath><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>∊</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>)</m:r></span></span></i></m:omath><span dir="LTR" style="font-size:9.0pt"> we have </span><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><span lang="FA" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>∊</m:r><m:r>τ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>, </m:r><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>∊</m:r><m:r>τ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></i></m:omath><span dir="LTR" style="font-size:9.0pt"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:17.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Submanifolds of statistical cosymplectic manifolds </span></span></b></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">We show that the manifold </span></span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:e></m:d></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is a statistical cosymplectic manifold if and only if </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>φ</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is a statistical cosymplectic manifold. Moreover we prove the following theorems. </span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Theorem. </span></span></b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Any invariant submanifold of a </span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">statistical cosymplectic manifold with tangent structure vector field </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">, is a statistical cosymplectic and minimal-like submanifold.</span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Theorem. </span></span></b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> be a submanifold of </span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">statistical cosymplectic manifold </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> such that the structure vector field </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is normal to </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">. Then for any vector field </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>∊</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> we have</span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omath><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>A</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></i></m:sup></m:ssup></m:e><m:sub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:sub></m:ssub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>=0, </m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e><m:sup><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>⊥</m:r></span></span></i></m:sup></m:ssup></m:e><m:sub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r><m:r>=</m:r><m:r>η</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span dir="LTR" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc></m:e><m:sub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r></span></span></i></m:sub></m:ssub><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:e></m:d><i><span dir="LTR" lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r><m:r>. </m:r></span></span></i></m:omath></m:omathpara><span lang="EN-GB" dir="LTR" style="font-size:9.0pt"></span></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Theorem. </span></span></b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:e></m:d></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> be a statistical cosymplectic manifold. If </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is a submanifold of </span></span><m:omath><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> and the structure vector field </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is normal to </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> then</span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><m:omathpara><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>⊥</m:r></span></span></i></m:sup></m:ssup><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>U</m:r><m:r>,</m:r><m:r>V</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r><m:r>=0</m:r></span></span></i><i><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>, ∀</m:r><m:r>U</m:r><m:r>,</m:r><m:r>V</m:r><m:r>∊</m:r><m:r>τ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:d><i><span lang="EN-GB" style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></i></m:omath></m:omathpara><span lang="EN-GB" style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Theorem. </span></span></b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> be an invariant submanifold of </span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">statistical cosymplectic manifold </span></span><m:omath><m:d><m:dpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></i><m:acc><m:accpr><m:chr m:val="̅"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∇</m:r></span></span></m:e></m:acc><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>,</m:r><m:r>φ</m:r><m:r>,</m:r><m:r>ξ</m:r><m:r>,</m:r><m:r>η</m:r><m:r>,</m:r><m:r>g</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:e></m:d></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> and </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>ξ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is normal to </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">. Then </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> is a statistical Keahler-like manifold.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:17.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt">Conclusion</span></b></span></span></span><br>
<span style="font-size:12pt"><span style="line-height:17.0pt"><span new="" roman="" style="font-family:" times=""><span style="font-size:9.0pt">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</span><span style="font-size:9.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span>
منیفلد آماری, ساختار دو-همتافته, منیفلد کیلری آماری
Statistical manifold, cosymplectic structure, statistical Keahlerian manifold
203
215
http://mmr.khu.ac.ir/browse.php?a_code=A-10-607-1&slc_lang=fa&sid=1
Mohammad Bagher
Kazemi
محمدباقر
کاظمی
mbkazemi@znu.ac.ir
10031947532846005479
10031947532846005479
Yes
دانشگاه زنجان
Shiva
Salahvarzi
شیوا
سلاح ورزی
s.salahvarzi@znu.ac.ir
10031947532846005480
10031947532846005480
No
دانشگاه زنجان