Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

Bibliographic Details
Main Author: Elshafei, Ahmed
Publication Date: 2023
Other Authors: Ferreira, Ana Cristina, Reis, Helena
Format: Article
Language: eng
Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
Download full: https://hdl.handle.net/1822/83717
Summary: This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2, R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2, ℂ).
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spelling Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groupsGeodesic (semi)completenessEuler-Arnold equationsHolomorphic metricCiências Naturais::MatemáticasThis paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2, R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2, ℂ).The first author was financed by FCT - Fundação para a Ciência e Tecnologia, I.P. (Portugal) - through the PhD scholarship PD/BD/143019/2018. The second author was partially supported by FCT, Portugal through the sabbatical grant SFRH/BSAB/135549/2018 and through CMAT, Portugal under the project UID/MAT/00013/2013. The third author was partially supported by CMUP, Portugal, member of LASI, which is financed by national funds through FCT under the project UIDB/00144/2020 and also by CIMI, France through the project “Complex dynamics of group actions, Halphen and Painlevé systems”. Finally, all three authors benefited from CNRS (France) support through the PICS project “Dynamics of Complex ODEs and Geometry”.ElsevierUniversidade do MinhoElshafei, AhmedFerreira, Ana CristinaReis, Helena2023-022023-02-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/83717eng0362-546X1873-521510.1016/j.na.2023.113252113252https://www.sciencedirect.com/science/article/pii/S0362546X23000445info:eu-repo/semantics/openAccessreponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiainstacron:RCAAP2025-04-12T04:24:14Zoai:repositorium.sdum.uminho.pt:1822/83717Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireinfo@rcaap.ptopendoar:https://opendoar.ac.uk/repository/71602025-05-28T15:07:26.973168Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiafalse
dc.title.none.fl_str_mv Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
title Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
spellingShingle Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
Elshafei, Ahmed
Geodesic (semi)completeness
Euler-Arnold equations
Holomorphic metric
Ciências Naturais::Matemáticas
title_short Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
title_full Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
title_fullStr Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
title_full_unstemmed Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
title_sort Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups
author Elshafei, Ahmed
author_facet Elshafei, Ahmed
Ferreira, Ana Cristina
Reis, Helena
author_role author
author2 Ferreira, Ana Cristina
Reis, Helena
author2_role author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Elshafei, Ahmed
Ferreira, Ana Cristina
Reis, Helena
dc.subject.por.fl_str_mv Geodesic (semi)completeness
Euler-Arnold equations
Holomorphic metric
Ciências Naturais::Matemáticas
topic Geodesic (semi)completeness
Euler-Arnold equations
Holomorphic metric
Ciências Naturais::Matemáticas
description This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2, R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2, ℂ).
publishDate 2023
dc.date.none.fl_str_mv 2023-02
2023-02-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://hdl.handle.net/1822/83717
url https://hdl.handle.net/1822/83717
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 0362-546X
1873-5215
10.1016/j.na.2023.113252
113252
https://www.sciencedirect.com/science/article/pii/S0362546X23000445
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron:RCAAP
instname_str FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron_str RCAAP
institution RCAAP
reponame_str Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
collection Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
repository.name.fl_str_mv Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
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