Left-Invariant Einstein-like Metrics on Compact Lie Groups
In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We pr...
Saved in:
| Published in | Mathematics (Basel) Vol. 10; no. 9; p. 1510 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Basel
MDPI AG
01.05.2022
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We prove that the left metric ⟨·,·⟩t1,t2 on G defined by the first equation, must be an A-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B-metrics, such as ⟨·,·⟩t1,t2. |
|---|---|
| AbstractList | In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G , such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We prove that the left metric ⟨·,·⟩t1,t2 on G defined by the first equation, must be an A -metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B -metrics, such as ⟨·,·⟩t1,t2 . In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G/H has exactly two inequivalent, irreducible summands. We prove that the left metric 〈·,·〉t1,t2 on G defined by the first equation, must be an A-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B-metrics, such as 〈·,·〉t1,t2. |
| Author | Wu, An Sun, Huafei |
| Author_xml | – sequence: 1 givenname: An surname: Wu fullname: Wu, An – sequence: 2 givenname: Huafei surname: Sun fullname: Sun, Huafei |
| BookMark | eNpNkM1OwzAQhC1UJErpjQeIxJWA7U2chBuqSqkUxAXO1tY_4NLGwXaReHtSilBPu9r5NDuaczLqfGcIuWT0BqCht1tM74zShpWMnpAx57zKq0EYHe1nZBrjmtI9BnXRjMlda2zKl90XBoddyuaui8m4Lt-4D5M9mRScipnvspnf9qhS1jqTLYLf9fGCnFrcRDP9mxPy-jB_mT3m7fNiObtvcwW8SHkJioMCxRrQ1AgBnHGtS4uaAyphygKhXBlW61oois1K4IpViGCtRlEBTMjy4Ks9rmUf3BbDt_To5O_BhzeJITm1MZLZApVWAmuuCgM1gsLaNlrwpirrxg5eVwevPvjPnYlJrv0udEN8yYdolFdFzQfq-kCp4GMMxv5_ZVTuy5bHZcMPUThyxg |
| Cites_doi | 10.1007/BFb0087534 10.1016/j.difgeo.2015.10.001 10.1007/s10455-008-9109-9 10.1007/BF02394607 10.1007/BF00151525 10.1007/s002220050355 10.1090/tran/7025 10.1016/j.geomphys.2022.104517 10.1090/stml/022 10.1007/BF01388738 10.1007/978-0-387-49158-5 10.1017/CBO9780511807497 10.1007/s11425-019-9744-0 10.1016/j.aim.2015.09.003 10.1007/s00006-007-0050-y 10.1007/978-3-540-74311-8 10.1007/s00006-006-0014-7 |
| ContentType | Journal Article |
| Copyright | 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| Copyright_xml | – notice: 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| DBID | AAYXX CITATION 3V. 7SC 7TB 7XB 8AL 8FD 8FE 8FG 8FK ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO FR3 GNUQQ HCIFZ JQ2 K7- KR7 L6V L7M L~C L~D M0N M7S P62 PHGZM PHGZT PIMPY PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U DOA |
| DOI | 10.3390/math10091510 |
| DatabaseName | CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts ProQuest Central (purchase pre-March 2016) Computing Database (Alumni Edition) Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Central Technology Collection ProQuest One ProQuest Central Korea Engineering Research Database ProQuest Central Student SciTech Premium Collection ProQuest Computer Science Collection Computer Science Database Civil Engineering Abstracts ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Engineering Database ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Premium ProQuest One Academic Publicly Available Content Database ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering Collection ProQuest Central Basic DOAJ Directory of Open Access Journals |
| DatabaseTitle | CrossRef Publicly Available Content Database Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection Civil Engineering Abstracts ProQuest Computing Engineering Database ProQuest Central Basic ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional ProQuest One Academic UKI Edition Materials Science & Engineering Collection Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ProQuest Central (Alumni) |
| DatabaseTitleList | CrossRef Publicly Available Content Database |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website – sequence: 2 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 2227-7390 |
| ExternalDocumentID | oai_doaj_org_article_1f4acdc6a82c4e38a3ca8f9d6297589f 10_3390_math10091510 |
| GroupedDBID | -~X 5VS 85S 8FE 8FG AADQD AAFWJ AAYXX ABDBF ABJCF ABPPZ ABUWG ACIPV ACIWK ADBBV AFKRA AFZYC ALMA_UNASSIGNED_HOLDINGS AMVHM ARAPS AZQEC BCNDV BENPR BGLVJ BPHCQ CCPQU CITATION DWQXO GNUQQ GROUPED_DOAJ HCIFZ IAO ITC K6V K7- KQ8 L6V M7S MODMG M~E OK1 PHGZM PHGZT PIMPY PQQKQ PROAC PTHSS RNS 3V. 7SC 7TB 7XB 8AL 8FD 8FK FR3 JQ2 KR7 L7M L~C L~D M0N P62 PKEHL PQEST PQGLB PQUKI PRINS Q9U PUEGO |
| ID | FETCH-LOGICAL-c324t-53c23c3c193d0e663212dd5fad23ac6e54a35be18d86c0a9b6ab17aa3ffda6733 |
| IEDL.DBID | BENPR |
| ISSN | 2227-7390 |
| IngestDate | Wed Aug 27 01:29:08 EDT 2025 Fri Jul 25 12:00:46 EDT 2025 Tue Jul 01 02:58:11 EDT 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 9 |
| Language | English |
| License | https://creativecommons.org/licenses/by/4.0 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c324t-53c23c3c193d0e663212dd5fad23ac6e54a35be18d86c0a9b6ab17aa3ffda6733 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| OpenAccessLink | https://www.proquest.com/docview/2663027482?pq-origsite=%requestingapplication% |
| PQID | 2663027482 |
| PQPubID | 2032364 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_1f4acdc6a82c4e38a3ca8f9d6297589f proquest_journals_2663027482 crossref_primary_10_3390_math10091510 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2022-05-01 |
| PublicationDateYYYYMMDD | 2022-05-01 |
| PublicationDate_xml | – month: 05 year: 2022 text: 2022-05-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Basel |
| PublicationPlace_xml | – name: Basel |
| PublicationTitle | Mathematics (Basel) |
| PublicationYear | 2022 |
| Publisher | MDPI AG |
| Publisher_xml | – name: MDPI AG |
| References | ref_12 Cafaro (ref_25) 2007; 17 Gray (ref_9) 1978; 7 ref_19 Qian (ref_13) 2021; 64 Peng (ref_11) 2016; 44 ref_16 Wang (ref_2) 2012; 22 Ziller (ref_15) 1979; 18 Dynkin (ref_6) 1957; 6 Tang (ref_10) 2015; 285 Wang (ref_18) 1986; 84 Dynkin (ref_5) 1957; 6 Kerr (ref_8) 2006; 8 ref_24 Wolf (ref_3) 1968; 120 ref_23 ref_22 Cafaro (ref_26) 2007; 17 ref_21 ref_20 (ref_17) 1999; 138 (ref_7) 1975; 8 ref_1 Yan (ref_14) 2017; 369 Dickinson (ref_4) 2008; 34 |
| References_xml | – volume: 18 start-page: 1 year: 1979 ident: ref_15 article-title: Naturally reductive metrics and Einstein metrics on compact Lie groups publication-title: Men. Amer. Math. Soc. – ident: ref_22 doi: 10.1007/BFb0087534 – volume: 8 start-page: 691 year: 1975 ident: ref_7 article-title: Eine Klassifikation bestimmter Untergruppen Kompakter Zusammenhängender Liegruppen publication-title: Commun. Algebra – volume: 44 start-page: 63 year: 2016 ident: ref_11 article-title: Homogeneous Einstein-like metrics on spheres and projective spaces publication-title: Diff. Geom. Appl. doi: 10.1016/j.difgeo.2015.10.001 – volume: 22 start-page: 253 year: 2012 ident: ref_2 article-title: Einstein metrics from symmetry and bundle constructions: A sequel publication-title: Adv. Lect. Math. – volume: 34 start-page: 329 year: 2008 ident: ref_4 article-title: The geometry of compact homogeneous spaces with two isotropy summands publication-title: Ann. Glob. Anal. Geom. doi: 10.1007/s10455-008-9109-9 – volume: 120 start-page: 59 year: 1968 ident: ref_3 article-title: The geometry and structure of isotropy irreducible homogeneous spaces publication-title: Acta. Math. doi: 10.1007/BF02394607 – volume: 7 start-page: 259 year: 1978 ident: ref_9 article-title: Einstein-like manifolds which are not Einstein publication-title: Geom. Dedic. doi: 10.1007/BF00151525 – volume: 8 start-page: 1455 year: 2006 ident: ref_8 article-title: Low-dimensional homogeneous Einstein manifolds publication-title: Trans. Amer. Math. Soc. – ident: ref_23 – ident: ref_21 – volume: 6 start-page: 245 year: 1957 ident: ref_6 article-title: Maximal subgroups of the classical groups publication-title: Amer. Math. Soc. Transl. Ser. 2 – volume: 138 start-page: 631 year: 1999 ident: ref_17 article-title: Optimal pinching constants of odd dimensional homogeneous spaces publication-title: Invent. Math. doi: 10.1007/s002220050355 – volume: 369 start-page: 8587 year: 2017 ident: ref_14 article-title: Einstien metrics on compact simple Lie groups attached to standard triples publication-title: Trans. Amer. Math. Soc. doi: 10.1090/tran/7025 – volume: 6 start-page: 111 year: 1957 ident: ref_5 article-title: Semisimple subalgebras of semisimple Lie algebras publication-title: Amer. Math. Soc. Transl. Ser. 2 – ident: ref_12 doi: 10.1016/j.geomphys.2022.104517 – ident: ref_16 doi: 10.1090/stml/022 – volume: 84 start-page: 177 year: 1986 ident: ref_18 article-title: Existence and non-existence of homogeneous Einstein metrics publication-title: Invent. Math. doi: 10.1007/BF01388738 – ident: ref_20 doi: 10.1007/978-0-387-49158-5 – ident: ref_24 doi: 10.1017/CBO9780511807497 – volume: 64 start-page: 1045 year: 2021 ident: ref_13 article-title: Homogeneous Einstein-like metrics on symmetric spaces publication-title: Sci. China Math. doi: 10.1007/s11425-019-9744-0 – volume: 285 start-page: 1970 year: 2015 ident: ref_10 article-title: Isoparametric foliation and a problem of Besse on generalizations of Einstein condition publication-title: Adv. Math. doi: 10.1016/j.aim.2015.09.003 – ident: ref_19 – volume: 17 start-page: 617 year: 2007 ident: ref_26 article-title: Finite-range electromagnetic interaction and magnetic charges spacetime algebra or algebra of physical space? publication-title: Adv. Appl. Clifford Algebr. doi: 10.1007/s00006-007-0050-y – ident: ref_1 doi: 10.1007/978-3-540-74311-8 – volume: 17 start-page: 23 year: 2007 ident: ref_25 article-title: The spacetime algebra approach to massive classical electrodynamics with magnetic monopoles publication-title: Adv. Appl. Clifford Algebr. doi: 10.1007/s00006-006-0014-7 |
| SSID | ssj0000913849 |
| Score | 2.1890512 |
| Snippet | In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G, such that G/K is a... In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H⊂K⊂G , such that G/K is a... |
| SourceID | doaj proquest crossref |
| SourceType | Open Website Aggregation Database Index Database |
| StartPage | 1510 |
| SubjectTerms | ({\mathcal{A}}\)-metric compact Lie groups Decomposition Einstein-like metric homogeneous space Invariants Isotropy Lie groups Mathematics Subgroups ℬ-metric |
| SummonAdditionalLinks | – databaseName: DOAJ Directory of Open Access Journals dbid: DOA link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LS8NAEF6kJz2IT6xWyUGPodlnNt5UWqq0niz0FvaJQUnFRH-_s0laAh68eF02JHyzM_PNZh4IXQuVYuISGUudsZhxByplpY6x9Kn2hBlrwoX-4lnMluxpxVe9UV8hJ6xtD9wCN8aeKXhAKEkMc1QqapT0mRWhJFRmPlhf8Hm9YKqxwRmmkmVtpjuFuH4M_O8Vh3UeimV7Pqhp1f_LEjfuZXqA9jteGN2133OIdlx5hPYW26aq1TG6nTtfx4_lN4S3gEc0KYDauaKM34s3Fy3CaCxTResyanTc1NG8cFFzt1SdoOV08vIwi7vRB7EBhlPHnBpCDTVAr2zigBWAh7GWe2UJVUY4zhTl2mFppTCJyrRQGqdKUe-tEimlp2hQrkt3hiKHtcASYuBEMmZhk84It8JzlRAvlR-imw0Y-Ufb4SKHyCCAlvdBG6L7gNR2T-hL3SyAtPJOWvlf0hqi0QbnvFOWKgeOEP6eMknO_-MdF2iXhBqFJitxhAb155e7BOZQ66vmkPwAziPB5w priority: 102 providerName: Directory of Open Access Journals |
| Title | Left-Invariant Einstein-like Metrics on Compact Lie Groups |
| URI | https://www.proquest.com/docview/2663027482 https://doaj.org/article/1f4acdc6a82c4e38a3ca8f9d6297589f |
| Volume | 10 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1JT9tAFB5BcqEH1IWqLI18KMdR7NnyzKUqVUJASVQhkLhZsxYL5AA2_f194zhppEpcxyON_L1567yFkG9KjzLmU6BgckGF9MhSDgzNIIxMYMI6GwP684Wa3oqrO3nXBdzqLq1yLRNbQe2WNsbIh6hI4hObAPb96ZnGqVHxdbUbobFL-iiCAXqkfz5e_LreRFli10sQ-SrjnaN_P0Q78D6L6zIWzW7porZl_38SuVUzk_dkv7MPkx8rgn4gO776SN7NN81V60_kbOZDQy-rP-jmIi7JuEQTz5cVfSwffDKPI7JsnSyrpOV12ySz0idtjKk-ILeT8c3PKe1GIFCLlk5DJbeMW27RzHKpRwjwN52TQTvGtVVeCs2l8Rk4UDbVuVHaZCOteQhOqxHnn0mvWlb-C0l8ZlQG6AunIITDTSZn0qkgdcoC6HBITtdgFE-rThcFeggRtGIbtENyHpHa7In9qduF5cvvorvuRRaERjIrDcwKz0FzqyHkTsVCXsjxsJM1zkXHNHXxj8RHb38-JnssViG0eYcnpNe8vPqvaBs0ZkB2YXIx6K7BoPWw_wIba7um |
| linkProvider | ProQuest |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1JTxRBFK4QPCgHoiJhU-ogxwrdtU01iTGojDMwwwkSbk2tMNH0AN1o_FP-Rt7rmR5ITLxxrap0J6--t9ZbCPmobS_nMTPMuEIyqSKwVDCO5Sb1XOLSB48B_fGpHpzL4wt1sUT-drUwmFbZycRWUIepxxj5PigSfGKThn--uWU4NQpfV7sRGjNYnMQ_v8Flqz8Nv8H97nHePzr7OmDzqQLMg_HQMCU8F154sFxCFuGrILxDUMkGLqzXUUkrlIu5CUb7zBZOW5f3rBUpBat7GAAFkf9CCtDkWJne_76I6WCPTSOLWX497Gf7YHVe57iusET3ieZrBwT8I_9bpdZ_TVbn1ig9nMHnDVmK1VuyMl60cq3XyMEopoYNq1_gVMMt0KMJGJRxUrGfkx-RjnEgl6_ptKKtZPENHU0ibSNa9Tty_iykWSfL1bSKG4TG3OncgOedGSkDHHIFV0EnZTOejE2bZK8jRnkz66tRgj-CRCufEm2TfEFKLc5gN-x2YXp3Vc6Zq8yTtAAqbQ33MgpjhbcmFUFj2bAp4Gc7HZ3LOYvW5SOgtv6_vUteDs7Go3I0PD3ZJq841j-0GY87ZLm5u4_vwSpp3IcWCpRcPjf2HgBPofZT |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1baxQxFA5lC6IP4hWrVfNgH8PO5DYZQcTaXbp2dylioW9jrroos7UzKv41f53nzM6sBcG3viaBwMl3rjkXQl5oW-Q8ZoYZV0omVQSWCsax3KTCJS598BjQXyz18Zl8d67Od8jvoRYG0yoHmdgJ6rD2GCMfgyLBLzZp-Dj1aRGnR9PXF98YTpDCn9ZhnMYGIifx109w35pXsyN46wPOp5MPb49ZP2GAeTAkWqaE58ILD1ZMyCLcAII8BJVs4MJ6HZW0QrmYm2C0z2zptHV5Ya1IKVhdYDAUxP9ugV7RiOweTpan77cRHuy4aWS5ybYXoszGYIN-znFdYcHuFT3YjQv4Rxt0Km56h9zubVP6ZgOmu2Qn1vfIrcW2sWtzn7ycx9SyWf0DXGx4EzpZgXkZVzX7uvoS6QLHc_mGrmvayRnf0vkq0i6-1TwgZ9dCnIdkVK_r-IjQmDudG_DDMyNlgEOu5CropGzGk7FpjxwMxKguNl02KvBOkGjVVaLtkUOk1PYM9sbuFtaXn6qe1ao8SQsQ09ZwL6MwVnhrUhk0FhGbEi7bH-hc9QzbVH_h9fj_28_JDcBdNZ8tT56QmxyLIbr0x30yai-_x6dgorTuWY8FSj5eN_z-AJk7--U |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Left-Invariant+Einstein-like+Metrics+on+Compact+Lie+Groups&rft.jtitle=Mathematics+%28Basel%29&rft.au=Wu%2C+An&rft.au=Sun%2C+Huafei&rft.date=2022-05-01&rft.pub=MDPI+AG&rft.eissn=2227-7390&rft.volume=10&rft.issue=9&rft.spage=1510&rft_id=info:doi/10.3390%2Fmath10091510&rft.externalDBID=HAS_PDF_LINK |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2227-7390&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2227-7390&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2227-7390&client=summon |