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 Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 4, Pages 27–46 (Mi izv394)

On isometries of some Riemannian Lie groups

V. V. Gorbatsevich

Abstract: We study isometry groups of Lie groups endowed with left-invariant Riemannian metrics. We mainly consider triangular Lie groups. By the familiar Gordon–Wilson theorem, calculating the isometry groups of left-invariant metrics on such groups is reduced to calculating the automorphism groups of the corresponding Lie algebras and to distinguishing compact subgroups of these groups. We consider nilpotent Lie groups in more detail, with special attention to filiform Lie groups and their relatives (prefiliform, quasifiliform). As a rule, we state the main results in terms of the automorphism groups of Lie algebras and then give their geometric interpretation. Special attention is paid to finding the group of connected components of the isometry group (in particular, it is calculated for all filiform Lie groups) and to conditions guaranteeing that the group of rotations (that is, isometries preserving a given point) is finite for certain classes of Riemannian Lie groups.

DOI: https://doi.org/10.4213/im394

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English version:
Izvestiya: Mathematics, 2002, 66:4, 683–699

Bibliographic databases:

UDC: 519.46
MSC: 53C30, 17B40, 22E25

Citation: V. V. Gorbatsevich, “On isometries of some Riemannian Lie groups”, Izv. RAN. Ser. Mat., 66:4 (2002), 27–46; Izv. Math., 66:4 (2002), 683–699

Citation in format AMSBIB
\Bibitem{Gor02} \by V.~V.~Gorbatsevich \paper On isometries of some Riemannian Lie groups \jour Izv. RAN. Ser. Mat. \yr 2002 \vol 66 \issue 4 \pages 27--46 \mathnet{http://mi.mathnet.ru/izv394} \crossref{https://doi.org/10.4213/im394} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1942094} \zmath{https://zbmath.org/?q=an:1038.53048} \transl \jour Izv. Math. \yr 2002 \vol 66 \issue 4 \pages 683--699 \crossref{https://doi.org/10.1070/IM2002v066n04ABEH000394} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748509550} 

• http://mi.mathnet.ru/eng/izv394
• https://doi.org/10.4213/im394
• http://mi.mathnet.ru/eng/izv/v66/i4/p27

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This publication is cited in the following articles:
1. Zhiguang Hu, Shaoqiang Deng, “Three dimensional homogeneous Finsler manifolds”, Math. Nachr, 2012, n/a
2. Payne T.L., “Applications of Index Sets and Nikolayevsky Derivations To Positive Rank Nilpotent Lie Algebras”, J. Lie Theory, 24:1 (2014), 1–27
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