An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds

Hamid-Reza Fanaï; Atefeh Hasan-Zadeh

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 2, page 149-160
  • ISSN: 0862-7959

Abstract

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We study a problem of isometric compact 2-step nilmanifolds M / Γ using some information on their geodesic flows, where M is a simply connected 2-step nilpotent Lie group with a left invariant metric and Γ 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.

How to cite

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Fanaï, Hamid-Reza, and Hasan-Zadeh, Atefeh. "An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds." Mathematica Bohemica 144.2 (2019): 149-160. <http://eudml.org/doc/294662>.

@article{Fanaï2019,
abstract = {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.},
author = {Fanaï, Hamid-Reza, Hasan-Zadeh, Atefeh},
journal = {Mathematica Bohemica},
keywords = {nilpotent Lie group; isometric nilmanifolds; normalizer; Lie algebroid; normal subgroupoid system; inner automorphism},
language = {eng},
number = {2},
pages = {149-160},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds},
url = {http://eudml.org/doc/294662},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Fanaï, Hamid-Reza
AU - Hasan-Zadeh, Atefeh
TI - An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 2
SP - 149
EP - 160
AB - 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.
LA - eng
KW - nilpotent Lie group; isometric nilmanifolds; normalizer; Lie algebroid; normal subgroupoid system; inner automorphism
UR - http://eudml.org/doc/294662
ER -

References

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