An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds
We study a problem of isometric compact 2-step nilmanifolds $M/\Gamma$ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma$ is a cocompact discrete subgroup of isometries of $M$. Among various works conce...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2019-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/144/2/mb144_2_4.pdf |
| _version_ | 1818059612313616384 |
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| author | Hamid-Reza Fanaï Atefeh Hasan-Zadeh |
| author_facet | Hamid-Reza Fanaï Atefeh Hasan-Zadeh |
| author_sort | Hamid-Reza Fanaï |
| collection | DOAJ |
| description | We study a problem of isometric compact 2-step nilmanifolds $M/\Gamma$ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma$ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified. |
| first_indexed | 2024-12-10T13:19:17Z |
| format | Article |
| id | doaj.art-a0b023501908419a948eb4b78107516e |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-10T13:19:17Z |
| publishDate | 2019-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-a0b023501908419a948eb4b78107516e2022-12-22T01:47:24ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362019-07-01144214916010.21136/MB.2018.0041-17MB.2018.0041-17An application of Lie groupoids to a rigidity problem of 2-step nilmanifoldsHamid-Reza FanaïAtefeh Hasan-ZadehWe study a problem of isometric compact 2-step nilmanifolds $M/\Gamma$ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma$ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.http://mb.math.cas.cz/full/144/2/mb144_2_4.pdf nilpotent Lie group isometric nilmanifolds normalizer Lie algebroid normal subgroupoid system inner automorphism |
| spellingShingle | Hamid-Reza Fanaï Atefeh Hasan-Zadeh An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds Mathematica Bohemica nilpotent Lie group isometric nilmanifolds normalizer Lie algebroid normal subgroupoid system inner automorphism |
| title | An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds |
| title_full | An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds |
| title_fullStr | An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds |
| title_full_unstemmed | An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds |
| title_short | An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds |
| title_sort | application of lie groupoids to a rigidity problem of 2 step nilmanifolds |
| topic | nilpotent Lie group isometric nilmanifolds normalizer Lie algebroid normal subgroupoid system inner automorphism |
| url | http://mb.math.cas.cz/full/144/2/mb144_2_4.pdf |
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