Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric
We consider the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> connections <inline-formula><math xmlns="http://www....
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/11/1735 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849330962583781376 |
|---|---|
| author | Esmaeil Peyghan Leila Nourmohammadifar Ion Mihai |
| author_facet | Esmaeil Peyghan Leila Nourmohammadifar Ion Mihai |
| author_sort | Esmaeil Peyghan |
| collection | DOAJ |
| description | We consider the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> connections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup></semantics></math></inline-formula> on a statistical manifold <i>M</i> equipped with a pair of conjugate connections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mo>=</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>∇</mo><mo>*</mo></msup><mo>=</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow></semantics></math></inline-formula>, where the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> connection is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>+</mo><mi>λ</mi></mrow><mn>2</mn></mfrac></mstyle><mo>∇</mo><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>λ</mi></mrow><mn>2</mn></mfrac></mstyle><msup><mo>∇</mo><mo>*</mo></msup></mrow></semantics></math></inline-formula>. This paper develops expressions for the vertical and horizontal distributions on the tangent bundle of the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and introduces the concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-adapted frames. We also derive the Levi–Civita connection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mmultiscripts><mover accent="true"><mo>∇</mo><mo>^</mo></mover><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup></semantics></math></inline-formula> of the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, which is equipped with the Cheeger Gromoll-type metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>g</mi><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow></semantics></math></inline-formula>. We study the statistical structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mmultiscripts><mi>g</mi><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts><mo>,</mo><msup><mrow><mmultiscripts><mo>∇</mo><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, which is naturally induced from the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. By introducing a para-holomorphic structure on the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we construct a para-Hermitian structure on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> and examine its integrability. Finally, we present the conditions under which these bundles admit a para-holomorphic structure. |
| format | Article |
| id | doaj-art-8a512340c9f546788d5eea008bfd97f6 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-8a512340c9f546788d5eea008bfd97f62025-08-20T03:46:46ZengMDPI AGMathematics2227-73902025-05-011311173510.3390/math13111735Para-Holomorphic Statistical Structure with Cheeger Gromoll MetricEsmaeil Peyghan0Leila Nourmohammadifar1Ion Mihai2Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, IranDepartment of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, IranDepartment of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 010014 Bucharest, RomaniaWe consider the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> connections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup></semantics></math></inline-formula> on a statistical manifold <i>M</i> equipped with a pair of conjugate connections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∇</mo><mo>=</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>∇</mo><mo>*</mo></msup><mo>=</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow></semantics></math></inline-formula>, where the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> connection is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>+</mo><mi>λ</mi></mrow><mn>2</mn></mfrac></mstyle><mo>∇</mo><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>λ</mi></mrow><mn>2</mn></mfrac></mstyle><msup><mo>∇</mo><mo>*</mo></msup></mrow></semantics></math></inline-formula>. This paper develops expressions for the vertical and horizontal distributions on the tangent bundle of the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and introduces the concept of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-adapted frames. We also derive the Levi–Civita connection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mmultiscripts><mover accent="true"><mo>∇</mo><mo>^</mo></mover><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup></semantics></math></inline-formula> of the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, which is equipped with the Cheeger Gromoll-type metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>g</mi><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow></semantics></math></inline-formula>. We study the statistical structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mmultiscripts><mi>g</mi><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts><mo>,</mo><msup><mrow><mmultiscripts><mo>∇</mo><none></none><none></none><mprescripts></mprescripts><none></none><mrow><mi>C</mi><mi>G</mi></mrow></mmultiscripts></mrow><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula>, which is naturally induced from the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. By introducing a para-holomorphic structure on the statistical manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msup><mo>∇</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we construct a para-Hermitian structure on the tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>M</mi></mrow></semantics></math></inline-formula> and examine its integrability. Finally, we present the conditions under which these bundles admit a para-holomorphic structure.https://www.mdpi.com/2227-7390/13/11/1735statistical manifoldspara-holomorphic structuresCheeger Gromoll metric |
| spellingShingle | Esmaeil Peyghan Leila Nourmohammadifar Ion Mihai Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric Mathematics statistical manifolds para-holomorphic structures Cheeger Gromoll metric |
| title | Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric |
| title_full | Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric |
| title_fullStr | Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric |
| title_full_unstemmed | Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric |
| title_short | Para-Holomorphic Statistical Structure with Cheeger Gromoll Metric |
| title_sort | para holomorphic statistical structure with cheeger gromoll metric |
| topic | statistical manifolds para-holomorphic structures Cheeger Gromoll metric |
| url | https://www.mdpi.com/2227-7390/13/11/1735 |
| work_keys_str_mv | AT esmaeilpeyghan paraholomorphicstatisticalstructurewithcheegergromollmetric AT leilanourmohammadifar paraholomorphicstatisticalstructurewithcheegergromollmetric AT ionmihai paraholomorphicstatisticalstructurewithcheegergromollmetric |