{"title":"几乎阿贝尔李群上的调和g2结构","authors":"Andrés J. Moreno","doi":"10.1016/j.difgeo.2023.102060","DOIUrl":null,"url":null,"abstract":"<div><p>We consider left-invariant <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket <em>A</em> of the corresponding Lie algebra. In those terms, we establish the algebraic condition on <em>A</em> for each of the possible 16-torsion classes of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structure. In particular, we show that four of those torsion classes are not admissible, since <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> implies <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Finally, we use the above results to provide the algebraic criteria on <em>A</em>, satisfying the harmonic condition <span><math><mi>div</mi><mspace></mspace><mi>T</mi><mo>=</mo><mn>0</mn></math></span>, and this allows to identify which torsion classes are harmonic.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Harmonic G2-structures on almost Abelian Lie groups\",\"authors\":\"Andrés J. Moreno\",\"doi\":\"10.1016/j.difgeo.2023.102060\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider left-invariant <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket <em>A</em> of the corresponding Lie algebra. In those terms, we establish the algebraic condition on <em>A</em> for each of the possible 16-torsion classes of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structure. In particular, we show that four of those torsion classes are not admissible, since <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> implies <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span>. Finally, we use the above results to provide the algebraic criteria on <em>A</em>, satisfying the harmonic condition <span><math><mi>div</mi><mspace></mspace><mi>T</mi><mo>=</mo><mn>0</mn></math></span>, and this allows to identify which torsion classes are harmonic.</p></div>\",\"PeriodicalId\":51010,\"journal\":{\"name\":\"Differential Geometry and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-09-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224523000864\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224523000864","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Harmonic G2-structures on almost Abelian Lie groups
We consider left-invariant -structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket A of the corresponding Lie algebra. In those terms, we establish the algebraic condition on A for each of the possible 16-torsion classes of a -structure. In particular, we show that four of those torsion classes are not admissible, since implies . Finally, we use the above results to provide the algebraic criteria on A, satisfying the harmonic condition , and this allows to identify which torsion classes are harmonic.
期刊介绍:
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.