{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T15:00:27Z","timestamp":1775833227228,"version":"3.50.1"},"reference-count":6,"publisher":"American Mathematical Society (AMS)","issue":"7","license":[{"start":{"date-parts":[[2000,11,29]],"date-time":"2000-11-29T00:00:00Z","timestamp":975456000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r Subscript k Baseline left-parenthesis upper G right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r_k(G)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the set of all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    -roots of the identity in 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 show that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r Subscript k Baseline left-parenthesis upper G right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r_k(G)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is always an embedded submanifold 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                    , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    -symmetric spaces and we show, more generally, that every\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    -symmetric space of 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                    is a covering manifold of an embedded submanifold\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O r b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Orb<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    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                    . We compute also the Hessian of the inclusions of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r Subscript k Baseline left-parenthesis upper G right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>r<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r_k(G)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O r b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Orb<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    into\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                    , relative to the natural connection on the domain and to the symmetric connection on\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                    .\n                  <\/p>","DOI":"10.1090\/s0002-9939-99-05240-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"2181-2186","source":"Crossref","is-referenced-by-count":1,"special_numbering":"493","title":["Higher order symmetric spaces and the roots of the identity in a Lie group"],"prefix":"10.1090","volume":"128","author":[{"given":"Cec\u00edlia","family":"Ferreira","sequence":"first","affiliation":[]},{"given":"Armando","family":"Machado","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1999,11,29]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1515\/crll.1995.469.149","article-title":"Harmonic tori in spheres and complex projective spaces","volume":"469","author":"Burstall, F. E.","year":"1995","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"2","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095561","volume-title":"Twistor theory for Riemannian symmetric spaces","volume":"1424","author":"Burstall, Francis E.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540526021"},{"issue":"3","key":"3","first-page":"575","article-title":"Embedding flag manifolds of a Hermitian space \ud835\udc38 into the unitary group \ud835\udc48(\ud835\udc38)","volume":"7","author":"Ferreira, Cec\u00edlia","year":"1993","journal-title":"Boll. Un. Mat. Ital. B (7)"},{"key":"4","unstructured":"C. Ferreira & A. Machado: Some embeddings of the space of partially complex structures. Portugal. Math. 55 (1998), 485\u2013504."},{"key":"5","first-page":"343","article-title":"Riemannian manifolds with geodesic symmetries of order 3","volume":"7","author":"Gray, Alfred","year":"1972","journal-title":"J. Differential Geometry","ISSN":"https:\/\/id.crossref.org\/issn\/0022-040X","issn-type":"print"},{"issue":"1","key":"6","first-page":"1","article-title":"Harmonic maps into Lie groups: classical solutions of the chiral model","volume":"30","author":"Uhlenbeck, Karen","year":"1989","journal-title":"J. Differential Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-040X","issn-type":"print"}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/2000-128-07\/S0002-9939-99-05240-5\/S0002-9939-99-05240-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/2000-128-07\/S0002-9939-99-05240-5\/S0002-9939-99-05240-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:15:50Z","timestamp":1775830550000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/2000-128-07\/S0002-9939-99-05240-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,11,29]]},"references-count":6,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2000,7]]}},"alternative-id":["S0002-9939-99-05240-5"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-99-05240-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[1999,11,29]]}}}