Musical Isomorphisms and Statistical Manifolds
Let ( M , g , ∇ ( α ) ) be a statistical manifold and g ♭ : T M → T ∗ M be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the α -vertical and α -horizontal lifts on the tangent bundle of the statistical manifold M , we construct the g - α -vertical and g - α -horizont...
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| Veröffentlicht in: | Mediterranean journal of mathematics 2022-10, Vol.19 (5), Article 225 |
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| creator | Peyghan, Esmaeil Nourmohammadifar, Leila Uddin, Siraj |
| description | Let
(
M
,
g
,
∇
(
α
)
)
be a statistical manifold and
g
♭
:
T
M
→
T
∗
M
be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the
α
-vertical and
α
-horizontal lifts on the tangent bundle of the statistical manifold
M
, we construct the
g
-
α
-vertical and
g
-
α
-horizontal lifts on the cotangent bundle with the aid of the musical isomorphism
g
♭
. We prove that the Lie bracket of the
α
-horizontal lifts of vector fields to tangent and cotangent bundles is
g
♭
-related if and only if the
α
-curvature tensor is an even function of
α
. Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen
(
α
)
-connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study. |
| doi_str_mv | 10.1007/s00009-022-02141-z |
| format | Article |
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(
M
,
g
,
∇
(
α
)
)
be a statistical manifold and
g
♭
:
T
M
→
T
∗
M
be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the
α
-vertical and
α
-horizontal lifts on the tangent bundle of the statistical manifold
M
, we construct the
g
-
α
-vertical and
g
-
α
-horizontal lifts on the cotangent bundle with the aid of the musical isomorphism
g
♭
. We prove that the Lie bracket of the
α
-horizontal lifts of vector fields to tangent and cotangent bundles is
g
♭
-related if and only if the
α
-curvature tensor is an even function of
α
. Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen
(
α
)
-connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study.</description><identifier>ISSN: 1660-5446</identifier><identifier>EISSN: 1660-5454</identifier><identifier>DOI: 10.1007/s00009-022-02141-z</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Fields (mathematics) ; Isomorphism ; Lifts ; Manifolds ; Mathematics ; Mathematics and Statistics ; Tensors</subject><ispartof>Mediterranean journal of mathematics, 2022-10, Vol.19 (5), Article 225</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c249t-5cef0fe7150b9a32896d4da5a6d1a0ea38887b2752b476d1140c68c35b54eb1f3</citedby><cites>FETCH-LOGICAL-c249t-5cef0fe7150b9a32896d4da5a6d1a0ea38887b2752b476d1140c68c35b54eb1f3</cites><orcidid>0000-0002-2713-6253</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00009-022-02141-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00009-022-02141-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Peyghan, Esmaeil</creatorcontrib><creatorcontrib>Nourmohammadifar, Leila</creatorcontrib><creatorcontrib>Uddin, Siraj</creatorcontrib><title>Musical Isomorphisms and Statistical Manifolds</title><title>Mediterranean journal of mathematics</title><addtitle>Mediterr. J. Math</addtitle><description>Let
(
M
,
g
,
∇
(
α
)
)
be a statistical manifold and
g
♭
:
T
M
→
T
∗
M
be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the
α
-vertical and
α
-horizontal lifts on the tangent bundle of the statistical manifold
M
, we construct the
g
-
α
-vertical and
g
-
α
-horizontal lifts on the cotangent bundle with the aid of the musical isomorphism
g
♭
. We prove that the Lie bracket of the
α
-horizontal lifts of vector fields to tangent and cotangent bundles is
g
♭
-related if and only if the
α
-curvature tensor is an even function of
α
. Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen
(
α
)
-connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study.</description><subject>Fields (mathematics)</subject><subject>Isomorphism</subject><subject>Lifts</subject><subject>Manifolds</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Tensors</subject><issn>1660-5446</issn><issn>1660-5454</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kE1Lw0AQhhdRsFb_gKeA562zn0mOUtQWWjyo52Wz2WhKPupOcrC_3rURvTkwzDDzvjPwEHLNYMEA0luEGDkFzmMyyejhhMyY1kCVVPL0t5f6nFwg7gB4zgSfkcV2xNrZJllj3_Zh_15ji4ntyuR5sEONw3G5tV1d9U2Jl-Sssg36q586J68P9y_LFd08Pa6XdxvquMwHqpyvoPIpU1DkVvAs16UsrbK6ZBa8FVmWpQVPFS9kGmdMgtOZE6pQ0hesEnNyM93dh_5j9DiYXT-GLr40PGVcaZWDjio-qVzoEYOvzD7UrQ2fhoH55mImLiZyMUcu5hBNYjJhFHdvPvyd_sf1BYGeZQ0</recordid><startdate>20221001</startdate><enddate>20221001</enddate><creator>Peyghan, Esmaeil</creator><creator>Nourmohammadifar, Leila</creator><creator>Uddin, Siraj</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2713-6253</orcidid></search><sort><creationdate>20221001</creationdate><title>Musical Isomorphisms and Statistical Manifolds</title><author>Peyghan, Esmaeil ; Nourmohammadifar, Leila ; Uddin, Siraj</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c249t-5cef0fe7150b9a32896d4da5a6d1a0ea38887b2752b476d1140c68c35b54eb1f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Fields (mathematics)</topic><topic>Isomorphism</topic><topic>Lifts</topic><topic>Manifolds</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Tensors</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Peyghan, Esmaeil</creatorcontrib><creatorcontrib>Nourmohammadifar, Leila</creatorcontrib><creatorcontrib>Uddin, Siraj</creatorcontrib><collection>CrossRef</collection><jtitle>Mediterranean journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Peyghan, Esmaeil</au><au>Nourmohammadifar, Leila</au><au>Uddin, Siraj</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Musical Isomorphisms and Statistical Manifolds</atitle><jtitle>Mediterranean journal of mathematics</jtitle><stitle>Mediterr. J. Math</stitle><date>2022-10-01</date><risdate>2022</risdate><volume>19</volume><issue>5</issue><artnum>225</artnum><issn>1660-5446</issn><eissn>1660-5454</eissn><abstract>Let
(
M
,
g
,
∇
(
α
)
)
be a statistical manifold and
g
♭
:
T
M
→
T
∗
M
be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the
α
-vertical and
α
-horizontal lifts on the tangent bundle of the statistical manifold
M
, we construct the
g
-
α
-vertical and
g
-
α
-horizontal lifts on the cotangent bundle with the aid of the musical isomorphism
g
♭
. We prove that the Lie bracket of the
α
-horizontal lifts of vector fields to tangent and cotangent bundles is
g
♭
-related if and only if the
α
-curvature tensor is an even function of
α
. Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen
(
α
)
-connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00009-022-02141-z</doi><orcidid>https://orcid.org/0000-0002-2713-6253</orcidid></addata></record> |
| fulltext | fulltext |
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| ispartof | Mediterranean journal of mathematics, 2022-10, Vol.19 (5), Article 225 |
| issn | 1660-5446 1660-5454 |
| language | eng |
| recordid | cdi_proquest_journals_2712565906 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Fields (mathematics) Isomorphism Lifts Manifolds Mathematics Mathematics and Statistics Tensors |
| title | Musical Isomorphisms and Statistical Manifolds |
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