{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T17:55:16Z","timestamp":1769018116945,"version":"3.49.0"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T00:00:00Z","timestamp":1673222400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T00:00:00Z","timestamp":1673222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100016245","name":"Matematisk-Naturvitenskapelige Fakultet, Universitetet i Bergen","doi-asserted-by":"publisher","award":["non"],"award-info":[{"award-number":["non"]}],"id":[{"id":"10.13039\/501100016245","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Geom Anal"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples.<\/jats:p>","DOI":"10.1007\/s12220-022-01179-5","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T17:03:54Z","timestamp":1673283834000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Symmetric Spaces Rolling on Flat Spaces"],"prefix":"10.1007","volume":"33","author":[{"given":"V.","family":"Jurdjevic","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2686-6657","authenticated-orcid":false,"given":"I.","family":"Markina","sequence":"additional","affiliation":[]},{"given":"F.","family":"Silva Leite","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,9]]},"reference":[{"key":"1179_CR1","doi-asserted-by":"crossref","unstructured":"Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint, vol.\u00a087 of Encyclopaedia of Mathematical Sciences, Control Theory and Optimization, II. Springer, Berlin (2004)","DOI":"10.1007\/978-3-662-06404-7"},{"key":"1179_CR2","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1007\/BF01232676","volume":"114","author":"RL Bryant","year":"1993","unstructured":"Bryant, R.L., Hsu, L.: Rigidity of integral curves of rank $$2$$ distributions. Invent. Math. 114, 435\u2013461 (1993)","journal-title":"Invent. Math."},{"key":"1179_CR3","doi-asserted-by":"crossref","unstructured":"Chitour, Y., Godoy\u00a0Molina, M., Kokkonen, P.: The rolling problem: overview and challenges. In: Geometric Control Theory and Sub-Riemannian Geometry, vol.\u00a05 of Springer INdAM Ser., pp.\u00a0103\u2013122. Springer, Cham (2014)","DOI":"10.1007\/978-3-319-02132-4_7"},{"key":"1179_CR4","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1016\/j.anihpc.2012.05.005","volume":"29","author":"Y Chitour","year":"2012","unstructured":"Chitour, Y., Kokkonen, P.: Rolling manifolds on space forms. Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire 29, 927\u2013954 (2012)","journal-title":"Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire"},{"key":"1179_CR5","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1098\/rspa.1970.0015","volume":"314","author":"CJS Clarke","year":"1970","unstructured":"Clarke, C.J.S.: On the global isometric embedding of pseudo-Riemannian manifolds. Proc. R. Soc. Lond. Ser. A 314, 417\u2013428 (1970)","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"1179_CR6","doi-asserted-by":"crossref","unstructured":"Crouch, P., Leite, F.S.: Rolling motions of pseudo-orthogonal groups. In: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp.\u00a07485\u20137491. IEEE (2012)","DOI":"10.1109\/CDC.2012.6426140"},{"key":"1179_CR7","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1137\/S0895479895290954","volume":"20","author":"A Edelman","year":"1999","unstructured":"Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303\u2013353 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1179_CR8","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s10883-012-9139-2","volume":"18","author":"M Godoy Molina","year":"2012","unstructured":"Godoy Molina, M., Grong, E., Markina, I., Silva Leite, F.: An intrinsic formulation of the problem on rolling manifolds. J. Dyn. Control Syst. 18, 181\u2013214 (2012)","journal-title":"J. Dyn. Control Syst."},{"key":"1179_CR9","unstructured":"Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, vol.\u00a080 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1978)"},{"key":"1179_CR10","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1080\/00207720802184717","volume":"39","author":"K H\u00fcper","year":"2008","unstructured":"H\u00fcper, K., Kleinsteuber, M., Silva Leite, F.: Rolling Stiefel manifolds. Int. J. Syst. Sci. 39, 881\u2013887 (2008)","journal-title":"Int. J. Syst. Sci."},{"key":"1179_CR11","doi-asserted-by":"crossref","unstructured":"H\u00fcper, K., Krakowski, K.A., Leite, F.S.: Rolling maps and nonlinear data. In: Handbook of Variational Methods for Nonlinear Geometric Data, pp. 577\u2013610. Springer, Cham (2020)","DOI":"10.1007\/978-3-030-31351-7_21"},{"key":"1179_CR12","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/s10883-007-9027-3","volume":"13","author":"K H\u00fcper","year":"2007","unstructured":"H\u00fcper, K., Silva Leite, F.: On the geometry of rolling and interpolation curves on $$S^n$$, $${\\rm SO}_n$$, and Grassmann manifolds. J. Dyn. Control Syst. 13, 467\u2013502 (2007)","journal-title":"J. Dyn. Control Syst."},{"key":"1179_CR13","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/BF00375605","volume":"124","author":"V Jurdjevic","year":"1993","unstructured":"Jurdjevic, V.: The geometry of the plate-ball problem. Arch. Rational Mech. Anal. 124, 305\u2013328 (1993)","journal-title":"Arch. Rational Mech. Anal."},{"key":"1179_CR14","volume-title":"Geometric Control Theory","author":"V Jurdjevic","year":"1997","unstructured":"Jurdjevic, V.: Geometric Control Theory, vol. 52. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)"},{"key":"1179_CR15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316286852","volume-title":"Optimal Control and Geometry: Integrable Systems","author":"V Jurdjevic","year":"2016","unstructured":"Jurdjevic, V.: Optimal Control and Geometry: Integrable Systems, vol. 154. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)"},{"key":"1179_CR16","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1017\/S0305004108001084","volume":"144","author":"V Jurdjevic","year":"2008","unstructured":"Jurdjevic, V., Zimmerman, J.: Rolling sphere problems on spaces of constant curvature. Math. Proc. Camb. Philos. Soc. 144, 729\u2013747 (2008)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"1179_CR17","doi-asserted-by":"crossref","unstructured":"Korolko, A., Leite, F.S.: Kinematics for rolling a lorentzian sphere. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp.\u00a06522\u20136527. IEEE (2011)","DOI":"10.1109\/CDC.2011.6160592"},{"key":"1179_CR18","doi-asserted-by":"publisher","first-page":"145","DOI":"10.3934\/jgm.2020016","volume":"13","author":"KA Krakowski","year":"2021","unstructured":"Krakowski, K.A., Machado, L., Leite, F.S.: A unifying approach for rolling symmetric spaces. J. Geom. Mech. 13, 145\u2013166 (2021)","journal-title":"J. Geom. Mech."},{"key":"1179_CR19","volume-title":"Riemannian Manifolds: An Introduction to Curvature","author":"JM Lee","year":"2006","unstructured":"Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature, vol. 176. Springer, New York (2006)"},{"key":"1179_CR20","doi-asserted-by":"crossref","unstructured":"Leite, F.S., Louro, F.: Sphere rolling on sphere: alternative approach to kinematics and constructive proof of controllability. In: Dynamics, Games and Science, vol.\u00a01 of CIM Ser. Math. Sci., pp.\u00a0341\u2013356. Springer, Cham (2015)","DOI":"10.1007\/978-3-319-16118-1_19"},{"key":"1179_CR21","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.4310\/CAG.2016.v24.n5.a7","volume":"24","author":"I Markina","year":"2016","unstructured":"Markina, I., Leite, F.S.: Introduction to the intrinsic rolling with indefinite metric. Commun. Anal. Geom. 24, 1085\u20131106 (2016)","journal-title":"Commun. Anal. Geom."},{"key":"1179_CR22","doi-asserted-by":"publisher","first-page":"105","DOI":"10.3934\/jgm.2021033","volume":"14","author":"A Marques","year":"2022","unstructured":"Marques, A., Leite, F.S.: Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces. J. Geom. Mech. 14, 105\u2013129 (2022)","journal-title":"J. Geom. Mech."},{"key":"1179_CR23","doi-asserted-by":"publisher","first-page":"623","DOI":"10.2748\/tmj\/1178229921","volume":"30","author":"K Nomizu","year":"1978","unstructured":"Nomizu, K.: Kinematics and differential geometry of submanifolds. Tohoku Math. J. 30, 623\u2013637 (1978)","journal-title":"Tohoku Math. J."},{"key":"1179_CR24","unstructured":"O\u2019Neill, B.: Semi-Riemannian geometry with applications to relativity, vol.\u00a0103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1983)"},{"key":"1179_CR25","unstructured":"Sharpe, R.W.: Differential Geometry. Cartan\u2019s Generalization of Klein\u2019s Erlangen Program, vol.\u00a0166 of Graduate Texts in Mathematics. Springer, New York (1997)"},{"key":"1179_CR26","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/s00498-004-0143-2","volume":"17","author":"JA Zimmerman","year":"2005","unstructured":"Zimmerman, J.A.: Optimal control of the sphere $$S^n$$ rolling on $$E^n$$. Math. Control Signals Syst. 17, 14\u201337 (2005)","journal-title":"Math. Control Signals Syst."}],"container-title":["The Journal of Geometric Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12220-022-01179-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12220-022-01179-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12220-022-01179-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T23:47:59Z","timestamp":1675727279000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12220-022-01179-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,9]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["1179"],"URL":"https:\/\/doi.org\/10.1007\/s12220-022-01179-5","relation":{},"ISSN":["1050-6926","1559-002X"],"issn-type":[{"value":"1050-6926","type":"print"},{"value":"1559-002X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,9]]},"assertion":[{"value":"30 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"94"}}