On low-dimensional manifolds with isometric SO0( p, q )-actions
Identifieur interne : 002A58 ( Istex/Corpus ); précédent : 002A57; suivant : 002A59On low-dimensional manifolds with isometric SO0( p, q )-actions
Auteurs : Gestur Lafsson ; Raul Quiroga-BarrancoSource :
- Transformation Groups [ 1083-4362 ] ; 2012-09-01.
Abstract
Abstract: Let G be a non-compact simple Lie group with Lie algebra $ \mathfrak{g} $ . Denote with m( $ \mathfrak{g} $ ) the dimension of the smallest non-trivial $ \mathfrak{g} $ -module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m( $ \mathfrak{g} $ ) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case $ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $ , providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either $ {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) $ or $ {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) $ .
Url:
DOI: 10.1007/s00031-012-9194-5
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Denote with m( $ \mathfrak{g} $ ) the dimension of the smallest non-trivial $ \mathfrak{g} $ -module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m( $ \mathfrak{g} $ ) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case $ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $ , providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either $ {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) $ or $ {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) $ .</div></front></TEI><istex><corpusName>springer-journals</corpusName><author><json:item><name>Gestur Ólafsson</name><affiliations><json:string>Department of Mathematics, 322 Lockett Hall, Louisiana State University, 70803, Baton Rouge, LA, USA</json:string><json:string>E-mail: olafsson@math.lsu.edu</json:string></affiliations></json:item><json:item><name>Raul Quiroga-Barranco</name><affiliations><json:string>Centro de Investigación en Matemáticas, Guanajuato, Apartado Postal 402, 36000, Guanajuato, Mexico</json:string><json:string>E-mail: quiroga@cimat.mx</json:string></affiliations></json:item></author><articleId><json:string>9194</json:string><json:string>s00031-012-9194-5</json:string></articleId><arkIstex>ark:/67375/VQC-4P92W78T-L</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: Let G be a non-compact simple Lie group with Lie algebra $ \mathfrak{g} $ . Denote with m( $ \mathfrak{g} $ ) the dimension of the smallest non-trivial $ \mathfrak{g} $ -module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m( $ \mathfrak{g} $ ) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case $ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $ , providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either $ {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) $ or $ {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) $ .</abstract><qualityIndicators><score>8.56</score><pdfWordCount>13278</pdfWordCount><pdfCharCount>57728</pdfCharCount><pdfVersion>1.4</pdfVersion><pdfPageCount>26</pdfPageCount><pdfPageSize>439.37 x 666.141 pts</pdfPageSize><refBibsNative>false</refBibsNative><abstractWordCount>130</abstractWordCount><abstractCharCount>789</abstractCharCount><keywordCount>0</keywordCount></qualityIndicators><title>On low-dimensional manifolds with isometric SO0( p, q )-actions</title><genre><json:string>research-article</json:string></genre><host><title>Transformation 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Denote with m( $ \mathfrak{g} $ ) the dimension of the smallest non-trivial $ \mathfrak{g} $ -module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m( $ \mathfrak{g} $ ) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case $ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $ , providing an explicit description of M when the bound is achieved. 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TocLevels="0"><ArticleID>9194</ArticleID><ArticleDOI>10.1007/s00031-012-9194-5</ArticleDOI><ArticleSequenceNumber>10</ArticleSequenceNumber><ArticleTitle Language="En">On low-dimensional manifolds with isometric SO<Subscript>0</Subscript>(<Emphasis Type="Italic">p, q</Emphasis>)-actions</ArticleTitle><ArticleFirstPage>835</ArticleFirstPage><ArticleLastPage>860</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2012</Year><Month>7</Month><Day>20</Day></RegistrationDate><Received><Year>2011</Year><Month>9</Month><Day>28</Day></Received><Accepted><Year>2012</Year><Month>6</Month><Day>7</Day></Accepted><OnlineDate><Year>2012</Year><Month>7</Month><Day>26</Day></OnlineDate></ArticleHistory><ArticleCopyright><CopyrightHolderName>Springer Science+Business Media, LLC</CopyrightHolderName><CopyrightYear>2012</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1" CorrespondingAffiliationID="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Gestur</GivenName><FamilyName>Ólafsson</FamilyName></AuthorName><Contact><Email>olafsson@math.lsu.edu</Email></Contact></Author><Author AffiliationIDS="Aff2" CorrespondingAffiliationID="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Raul</GivenName><FamilyName>Quiroga-Barranco</FamilyName></AuthorName><Contact><Email>quiroga@cimat.mx</Email></Contact></Author><Affiliation ID="Aff1"><OrgName>Department of Mathematics, 322 Lockett Hall, Louisiana State University</OrgName><OrgAddress><City>Baton Rouge</City><State>LA</State><Postcode>70803</Postcode><Country Code="US">USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Centro de Investigación en Matemáticas</OrgName><OrgAddress><Postbox>Apartado Postal 402</Postbox><Street>Guanajuato</Street><City>Guanajuato</City><Postcode>36000</Postcode><Country Code="MX">Mexico</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="All"><Heading>Abstract</Heading><Para>Let <Emphasis Type="Italic">G</Emphasis> be a non-compact simple Lie group with Lie algebra <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ \mathfrak{g} $</EquationSource></InlineEquation>. Denote with <Emphasis Type="Italic">m</Emphasis>(<InlineEquation ID="IEq2"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq2.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ \mathfrak{g} $</EquationSource></InlineEquation>) the dimension of the smallest non-trivial <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq3.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ \mathfrak{g} $</EquationSource></InlineEquation>-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold <Emphasis Type="Italic">M</Emphasis> it is observed that dim(<Emphasis Type="Italic">M</Emphasis>) ≥ dim(<Emphasis Type="Italic">G</Emphasis>) + <Emphasis Type="Italic">m</Emphasis>(<InlineEquation ID="IEq4"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq4.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ \mathfrak{g} $</EquationSource></InlineEquation>) when <Emphasis Type="Italic">M</Emphasis> admits an isometric <Emphasis Type="Italic">G</Emphasis>-action with a dense orbit. The Main Theorem considers the case <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq5.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $</EquationSource></InlineEquation>, providing an explicit description of <Emphasis Type="Italic">M</Emphasis> when the bound is achieved. In such a case, <Emphasis Type="Italic">M</Emphasis> is (up to a finite covering) the quotient by a lattice of either <InlineEquation ID="IEq6"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq6.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) $</EquationSource></InlineEquation> or <InlineEquation ID="IEq7"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="31_2012_9194_Article_IEq7.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$ {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) $</EquationSource></InlineEquation>.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>The first author was supported by DMS grants 0801010 and 1101337.</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>The second author was supported by Conacyt, Concyteg and SNI.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On low-dimensional manifolds with isometric SO0( p, q )-actions</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>On low-dimensional manifolds with isometric SO0(p, q)-actions</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Gestur</namePart><namePart type="family">Ólafsson</namePart><affiliation>Department of Mathematics, 322 Lockett Hall, Louisiana State University, 70803, Baton Rouge, LA, USA</affiliation><affiliation>E-mail: olafsson@math.lsu.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal" displayLabel="corresp"><namePart type="given">Raul</namePart><namePart type="family">Quiroga-Barranco</namePart><affiliation>Centro de Investigación en Matemáticas, Guanajuato, Apartado Postal 402, 36000, Guanajuato, Mexico</affiliation><affiliation>E-mail: quiroga@cimat.mx</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>SP Birkhäuser Verlag Boston</publisher><place><placeTerm type="text">Boston</placeTerm></place><dateCreated encoding="w3cdtf">2011-09-28</dateCreated><dateIssued encoding="w3cdtf">2012-09-01</dateIssued><dateIssued encoding="w3cdtf">2012</dateIssued><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: Let G be a non-compact simple Lie group with Lie algebra $ \mathfrak{g} $ . Denote with m( $ \mathfrak{g} $ ) the dimension of the smallest non-trivial $ \mathfrak{g} $ -module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold M it is observed that dim(M) ≥ dim(G) + m( $ \mathfrak{g} $ ) when M admits an isometric G-action with a dense orbit. The Main Theorem considers the case $ G = {\widetilde {\text{SO}}_0}\left( {p,q} \right) $ , providing an explicit description of M when the bound is achieved. In such a case, M is (up to a finite covering) the quotient by a lattice of either $ {\widetilde {\text{SO}}_0}\left( {p + 1,q} \right) $ or $ {\widetilde {\text{SO}}_0}\left( {p,q + 1} \right) $ .</abstract><relatedItem type="host"><titleInfo><title>Transformation Groups</title></titleInfo><titleInfo type="abbreviated"><title>Transformation Groups</title></titleInfo><genre type="journal" displayLabel="Non Standard Archive Journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>Springer</publisher><dateIssued encoding="w3cdtf">2012-08-03</dateIssued><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Algebra</topic><topic>Topological Groups, Lie Groups</topic></subject><identifier type="ISSN">1083-4362</identifier><identifier type="eISSN">1531-586X</identifier><identifier type="JournalID">31</identifier><identifier type="IssueArticleCount">11</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>2012</date><detail type="volume"><number>17</number><caption>vol.</caption></detail><detail type="issue"><number>3</number><caption>no.</caption></detail><extent unit="pages"><start>835</start><end>860</end></extent></part><recordInfo><recordOrigin>Springer Science+Business Media, LLC, 2012</recordOrigin></recordInfo></relatedItem><identifier type="istex">CE913EC8805283482450C39C2C145EE4A220BF06</identifier><identifier type="ark">ark:/67375/VQC-4P92W78T-L</identifier><identifier type="DOI">10.1007/s00031-012-9194-5</identifier><identifier type="ArticleID">9194</identifier><identifier type="ArticleID">s00031-012-9194-5</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer Science+Business Media, LLC, 2012</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer Science+Business Media, LLC, 2012</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/CE913EC8805283482450C39C2C145EE4A220BF06/metadata/json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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|area= BourbakiV1
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|clé= ISTEX:CE913EC8805283482450C39C2C145EE4A220BF06
|texte= On low-dimensional manifolds with isometric SO0( p, q )-actions
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