{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T20:02:31Z","timestamp":1776369751585,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"8","license":[{"start":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T00:00:00Z","timestamp":1750377600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia e a Tecnologia","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.54499\/UIDB\/00013\/2020"],"award-info":[{"award-number":["https:\/\/doi.org\/10.54499\/UIDB\/00013\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia e a Tecnologia","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.54499\/UIDP\/00013\/2020"],"award-info":[{"award-number":["https:\/\/doi.org\/10.54499\/UIDP\/00013\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia e a Tecnologia","doi-asserted-by":"publisher","award":["PID2020-116126GBI00"],"award-info":[{"award-number":["PID2020-116126GBI00"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia e a Tecnologia","doi-asserted-by":"publisher","award":["CEX2020- 001105-MCIN\/AEI\/10.13039\/501100011033"],"award-info":[{"award-number":["CEX2020- 001105-MCIN\/AEI\/10.13039\/501100011033"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00c3\u00b3n","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.54499\/UIDB\/00013\/2020"],"award-info":[{"award-number":["https:\/\/doi.org\/10.54499\/UIDB\/00013\/2020"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00c3\u00b3n","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.54499\/UIDP\/00013\/2020"],"award-info":[{"award-number":["https:\/\/doi.org\/10.54499\/UIDP\/00013\/2020"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00c3\u00b3n","doi-asserted-by":"publisher","award":["PID2020-116126GBI00"],"award-info":[{"award-number":["PID2020-116126GBI00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00c3\u00b3n","doi-asserted-by":"publisher","award":["CEX2020- 001105-MCIN\/AEI\/10.13039\/501100011033"],"award-info":[{"award-number":["CEX2020- 001105-MCIN\/AEI\/10.13039\/501100011033"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    For each left-invariant semi-Riemannian metric\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on a Lie group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we introduce the class of bi-Lipschitz Riemannian\n                    <italic>Clairaut<\/italic>\n                    metrics, whose completeness implies the completeness of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . When the adjoint representation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K left-normal-factor-semidirect-product Subscript rho Baseline double-struck upper R Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:msub>\n                              <mml:mo>\n                                \u22c9\n                                \n                              <\/mml:mo>\n                              <mml:mi>\n                                \u03c1\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K \\ltimes _\\rho \\mathbb {R}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the direct product of a compact and an abelian Lie group and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho left-parenthesis upper K right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c1\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho (K)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the absence of linear growth and suggest new questions.\n                  <\/p>","DOI":"10.1090\/tran\/9160","type":"journal-article","created":{"date-parts":[[2024,3,29]],"date-time":"2024-03-29T16:21:05Z","timestamp":1711729265000},"page":"5837-5862","source":"Crossref","is-referenced-by-count":1,"special_numbering":"1083","title":["Lie groups with all left-invariant semi-Riemannian metrics complete"],"prefix":"10.1090","volume":"377","author":[{"given":"Ahmed","family":"Elshafei","sequence":"first","affiliation":[]},{"given":"Ana Cristina","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"Miguel","family":"S\u00e1nchez","sequence":"additional","affiliation":[]},{"given":"Abdelghani","family":"Zeghib","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,6,20]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Foundations of mechanics","author":"Abraham, Ralph","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/080530102X"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"895","DOI":"10.1007\/s00006-016-0660-3","article-title":"Tangent Lie groups are Riemannian naturally reductive spaces","volume":"27","author":"Agricola, Ilka","year":"2017","journal-title":"Adv. Appl. Clifford Algebr.","ISSN":"https:\/\/id.crossref.org\/issn\/0188-7009","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","first-page":"319","DOI":"10.5802\/aif.233","article-title":"Sur la g\u00e9om\u00e9trie diff\u00e9rentielle des groupes de Lie de dimension infinie et ses applications \u00e0 l\u2019hydrodynamique des fluides parfaits","volume":"16","author":"Arnold, V.","year":"1966","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[B89a] N. Bourbaki, General topology, Chapters 1\u20134, Elements of Mathematics, Springer (1989)","DOI":"10.1007\/978-3-642-61703-4"},{"key":"5","series-title":"Elements of Mathematics (Berlin)","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61703-4","volume-title":"General topology. Chapters 5--10","author":"Bourbaki, Nicolas","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3540193723"},{"key":"6","series-title":"Elements of Mathematics (Berlin)","isbn-type":"print","volume-title":"Lie groups and Lie algebras. Chapters 1--3","author":"Bourbaki, Nicolas","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3540502181"},{"key":"7","doi-asserted-by":"publisher","first-page":"Paper 088, 13","DOI":"10.3842\/SIGMA.2008.088","article-title":"Geodesically complete Lorentzian metrics on some homogeneous 3 manifolds","volume":"4","author":"Bromberg, Shirley","year":"2008","journal-title":"SIGMA Symmetry Integrability Geom. Methods Appl."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s00205-012-0596-2","article-title":"Completeness of the trajectories of particles coupled to a general force field","volume":"208","author":"Candela, Anna Maria","year":"2013","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"4","key":"9","first-page":"371","article-title":"Sur la compl\u00e9tude des pseudo-m\u00e9triques invariantes a gauche sur les groupes de Lie nilpotents","volume":"52","author":"Guediri, M.","year":"1994","journal-title":"Rend. Sem. Mat. Univ. Politec. Torino","ISSN":"https:\/\/id.crossref.org\/issn\/0373-1243","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"391","DOI":"10.2969\/jmsj\/03730391","article-title":"Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds","volume":"37","author":"Kanai, Masahiko","year":"1985","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"1065","DOI":"10.1512\/iumj.1973.22.22089","article-title":"On completeness of homogeneous pseudo-riemannian manifolds","volume":"22","author":"Marsden, Jerrold","year":"1972","journal-title":"Indiana Univ. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2518","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/S0001-8708(76)80002-3","article-title":"Curvatures of left invariant metrics on Lie groups","volume":"21","author":"Milnor, John","year":"1976","journal-title":"Advances in Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"13","series-title":"Pure and Applied Mathematics","isbn-type":"print","volume-title":"Semi-Riemannian geometry","volume":"103","author":"O\u2019Neill, Barrett","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0125267401"},{"key":"14","series-title":"Springer Series in Soviet Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-74334-4","volume-title":"Lie groups and algebraic groups","author":"Onishchik, A. L.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540506144"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/BF01264047","article-title":"On completeness of certain families of semi-Riemannian manifolds","volume":"53","author":"Romero, Alfonso","year":"1994","journal-title":"Geom. Dedicata","ISSN":"https:\/\/id.crossref.org\/issn\/0046-5755","issn-type":"print"},{"key":"16","isbn-type":"print","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/978-1-4939-2441-7_15","article-title":"On the completeness of trajectories for some mechanical systems","author":"S\u00e1nchez, Miguel","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781493924400"},{"issue":"5","key":"17","doi-asserted-by":"publisher","first-page":"611","DOI":"10.37863\/umzh.v72i5.645","article-title":"Geodesic completeness of the left-invariant metrics on \u211d\u210d\u207f","volume":"72","author":"Vukmirovi\u0107, S.","year":"2020","journal-title":"Ukra\\\"{\\i}n. Mat. Zh.","ISSN":"https:\/\/id.crossref.org\/issn\/1027-3190","issn-type":"print"},{"key":"18","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-1799-0","volume-title":"Foundations of differentiable manifolds and Lie groups","volume":"94","author":"Warner, Frank W.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908943"}],"container-title":["Transactions of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/tran\/2024-377-08\/S0002-9947-2024-09160-2\/tran9160_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/tran\/2024-377-08\/S0002-9947-2024-09160-2\/S0002-9947-2024-09160-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T19:08:24Z","timestamp":1776366504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/tran\/2024-377-08\/S0002-9947-2024-09160-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,20]]},"references-count":18,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["S0002-9947-2024-09160-2"],"URL":"https:\/\/doi.org\/10.1090\/tran\/9160","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6850","0002-9947"],"issn-type":[{"value":"1088-6850","type":"electronic"},{"value":"0002-9947","type":"print"}],"subject":[],"published":{"date-parts":[[2024,6,20]]}}}