{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T16:21:54Z","timestamp":1764433314066,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study a theory of gravity of the form     f ( G )     where    G    is the Gauss\u2013Bonnet topological invariant without considering the standard Einstein\u2013Hilbert term as common in the literature, in arbitrary     ( d + 1 )     dimensions. The approach is motivated by the fact that, in particular conditions, the Ricci curvature scalar can be easily recovered and then a pure     f ( G )     gravity can be considered a further generalization of General Relativity like     f ( R )     gravity. Searching for Noether symmetries, we specify the functional forms invariant under point transformations in a static and spherically symmetric spacetime and, with the help of these symmetries, we find exact solutions showing that Gauss\u2013Bonnet gravity is significant without assuming the Ricci scalar in the action.<\/jats:p>","DOI":"10.3390\/sym12030372","type":"journal-article","created":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T03:24:20Z","timestamp":1583292260000},"page":"372","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":41,"title":["Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss\u2013Bonnet Gravity"],"prefix":"10.3390","volume":"12","author":[{"given":"Francesco","family":"Bajardi","sequence":"first","affiliation":[{"name":"Department of Physics \u201cE. Pancini\u201d, University of Naples \u201cFederico II\u201d, 80138 Naples, Italy"},{"name":"INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy"}]},{"given":"Konstantinos F.","family":"Dialektopoulos","sequence":"additional","affiliation":[{"name":"Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China"}]},{"given":"Salvatore","family":"Capozziello","sequence":"additional","affiliation":[{"name":"Department of Physics \u201cE. Pancini\u201d, University of Naples \u201cFederico II\u201d, 80138 Naples, Italy"},{"name":"INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 Napoli, Italy"},{"name":"Gran Sasso Science Institute, viale F. Crispi 7, I-67100 L\u2019Aquila, Italy"},{"name":"Physics Department, Tomsk State Pedagogical University, ul. Kievskaya, 60, Tomsk 634061, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"ref_1","unstructured":"Perivolaropoulos, L. (2008). Six Puzzles for LCDM Cosmology. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.dark.2016.02.001","article-title":"Beyond \u039bCDM: Problems, solutions, and the road ahead","volume":"12","author":"Bull","year":"2016","journal-title":"Phys. Dark Univ."},{"key":"ref_3","unstructured":"Barack, L., Cardoso, V., Nissanke, S., Sotiriou, T.P., Askar, A., and Belczynski, C. (2018). Black holes, gravitational waves and fundamental physics: A roadmap. arXiv."},{"key":"ref_4","unstructured":"Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, Inc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physrep.2011.09.003","article-title":"Extended Theories of Gravity","volume":"509","author":"Capozziello","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.physrep.2011.04.001","article-title":"Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models","volume":"505","author":"Nojiri","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.06.001","article-title":"Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution","volume":"692","author":"Nojiri","year":"2017","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified Gravity and Cosmology","volume":"513","author":"Clifton","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1142\/S0218271802002025","article-title":"Curvature quintessence","volume":"11","author":"Capozziello","year":"2002","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_10","first-page":"49","article-title":"A Bird\u2019s eye view of f(R)-gravity","volume":"3","author":"Capozziello","year":"2010","journal-title":"Open Astron. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"35","DOI":"10.15625\/0868-3166\/29\/1\/13192","article-title":"Testing the f(R)-theory of gravity","volume":"29","author":"Anh","year":"2019","journal-title":"Commun. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1140\/epjc\/s10052-018-6023-6","article-title":"Perturbative solutions of the f(R)-theory of gravity in a central gravitational field and some applications","volume":"78","author":"Anh","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/PhysRev.124.925","article-title":"Mach\u2019s principle and a relativistic theory of gravitation","volume":"124","author":"Brans","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF01807638","article-title":"Second-order scalar-tensor field equations in a four-dimensional space","volume":"10","author":"Horndeski","year":"1974","journal-title":"Int. J. Theor. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"084003","DOI":"10.1103\/PhysRevD.79.084003","article-title":"Covariant Galileon","volume":"79","author":"Deffayet","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1140\/epjc\/s10052-018-5939-1","article-title":"Classification of the Horndeski cosmologies via Noether Symmetries","volume":"78","author":"Capozziello","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1007\/JHEP02(2012)126","article-title":"Bimetric Gravity from Ghost-free Massive Gravity","volume":"1202","author":"Hassan","year":"2012","journal-title":"JHEP"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"231101","DOI":"10.1103\/PhysRevLett.106.231101","article-title":"Resummation of Massive Gravity","volume":"106","author":"Gabadadze","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"111301","DOI":"10.1103\/PhysRevLett.99.111301","article-title":"Nonlocal Cosmology","volume":"99","author":"Deser","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1730020","DOI":"10.1142\/S0218271817300208","article-title":"Nonlocal quantum gravity: A review","volume":"26","author":"Modesto","year":"2017","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124035","DOI":"10.1103\/PhysRevD.100.124035","article-title":"Cosmology of Lorentz fiber-bundle induced scalar-tensor theories","volume":"100","author":"Ikeda","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Aldrovandi, R., and Pereira, J.G. (2013). Teleparallel Gravity: An Introduction, Springer.","DOI":"10.1007\/978-94-007-5143-9"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"106901","DOI":"10.1088\/0034-4885\/79\/10\/106901","article-title":"f(T) teleparallel gravity and cosmology","volume":"79","author":"Cai","year":"2016","journal-title":"Rep. Prog. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1063\/1.1665613","article-title":"The Einstein tensor and its generalizations","volume":"12","author":"Lovelock","year":"1971","journal-title":"J. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/0927-6505(93)90017-8","article-title":"Pre - big bang in string cosmology","volume":"1","author":"Gasperini","year":"1993","journal-title":"Astropart. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physletb.2005.10.010","article-title":"Modified Gauss\u2013Bonnet theory as gravitational alternative for dark energy","volume":"631","author":"Nojiri","year":"2005","journal-title":"Phys. Lett. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physletb.2009.03.060","article-title":"Construction of cosmologically viable f(G) dark energy models","volume":"675","author":"Felice","year":"2009","journal-title":"Phys. Lett. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2725","DOI":"10.1007\/s10714-009-0803-0","article-title":"Cosmological scaling solutions in generalised Gauss\u2013Bonnet gravity theories","volume":"41","author":"Uddin","year":"2009","journal-title":"Gen. Relat. Grav."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"023532","DOI":"10.1103\/PhysRevD.81.023532","article-title":"Matter instabilities in general Gauss\u2013Bonnet gravity","volume":"81","author":"Felice","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Davis, S.C. (2007). Solar system constraints on f(G) dark energy. arXiv.","DOI":"10.1063\/1.2823806"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"225001","DOI":"10.1088\/0264-9381\/33\/22\/225001","article-title":"Spherically symmetric brane in a bulk of f(R) and Gauss\u2013Bonnet gravity","volume":"33","author":"Chakraborty","year":"2016","journal-title":"Class. Quant. Grav."},{"key":"ref_32","first-page":"087","article-title":"AdS(7)\/CFT(6), Gauss\u2013Bonnet Gravity, and Viscosity Bound","volume":"1003","author":"Kulaxizi","year":"2010","journal-title":"JHEP"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"185007","DOI":"10.1088\/0264-9381\/32\/18\/185007","article-title":"Modified Gauss - Bonnet gravity with the Lagrange multiplier constraint as mimetic theory","volume":"32","author":"Astashenok","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1450082","DOI":"10.1142\/S0219887814500820","article-title":"Newtonian, Post Newtonian and Parameterized Post Newtonian limits of f(R,G) gravity","volume":"11","author":"Laurentis","year":"2014","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/0370-2693(83)91253-4","article-title":"Do We Live Inside a Domain Wall?","volume":"125B","author":"Rubakov","year":"1983","journal-title":"Phys. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/0370-2693(90)90617-F","article-title":"A Possible new dimension at a few TeV","volume":"246","author":"Antoniadis","year":"1990","journal-title":"Phys. Lett. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3370","DOI":"10.1103\/PhysRevLett.83.3370","article-title":"A Large mass hierarchy from a small extra dimension","volume":"83","author":"Randall","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4690","DOI":"10.1103\/PhysRevLett.83.4690","article-title":"An Alternative to compactification","volume":"83","author":"Randall","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0370-2693(98)00466-3","article-title":"The Hierarchy problem and new dimensions at a millimeter","volume":"429","author":"Dimopoulos","year":"1998","journal-title":"Phys. Lett. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1840007","DOI":"10.1142\/S0219887818400078","article-title":"Noether Symmetries as a geometric criterion to select theories of gravity","volume":"15","author":"Dialektopoulos","year":"2018","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02742992","article-title":"Noether symmetries in cosmology","volume":"19","author":"Capozziello","year":"1996","journal-title":"Riv. Nuovo Cim."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Tsamparlis, M., and Paliathanasis, A. (2018). Symmetries of Differential Equations in Cosmology. Symmetry, 10.","DOI":"10.3390\/sym10070233"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"103526","DOI":"10.1103\/PhysRevD.88.103526","article-title":"Noether symmetries and analytical solutions in f(T)-cosmology: A complete study","volume":"88","author":"Basilakos","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1140\/epjc\/s10052-017-4677-0","article-title":"Noether Symmetry Approach in f(T,B) teleparallel cosmology","volume":"77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1140\/epjc\/s10052-016-4491-0","article-title":"Noether symmetries in Gauss\u2013Bonnet-teleparallel cosmology","volume":"76","author":"Capozziello","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1140\/epjc\/s10052-017-5283-x","article-title":"Constraining Generalized Non-local Cosmology from Noether Symmetries","volume":"77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1143\/PTP.126.511","article-title":"Generalized G-inflation: Inflation with the most general second-order field equations","volume":"126","author":"Kobayashi","year":"2011","journal-title":"Prog. Theor. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"195022","DOI":"10.1088\/0264-9381\/32\/19\/195022","article-title":"Birkhoff\u2019s theorem in Lovelock gravity for general base manifolds","volume":"32","author":"Ray","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"072502","DOI":"10.1063\/1.1960798","article-title":"Birkhoff\u2019s theorem in Lovelock gravity","volume":"46","author":"Zegers","year":"2005","journal-title":"J. Math. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"024002","DOI":"10.1103\/PhysRevD.82.024002","article-title":"Static solutions with nontrivial boundaries for the Einstein-Gauss\u2013Bonnet theory in vacuum","volume":"82","author":"Dotti","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1063\/1.3462680","article-title":"Extensions of Birkhoff\u2019s theorem in 6D Gauss\u2013Bonnet gravity","volume":"1241","author":"Bogdanos","year":"2010","journal-title":"AIP Conf. Proc."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"037","DOI":"10.1088\/1126-6708\/2009\/10\/037","article-title":"Einstein-Gauss\u2013Bonnet metrics: Black holes, black strings and a staticity theorem","volume":"0910","author":"Bogdanos","year":"2009","journal-title":"JHEP"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"104011","DOI":"10.1103\/PhysRevD.86.104011","article-title":"Critical Collapse in Einstein-Gauss\u2013Bonnet Gravity in Five and Six Dimensions","volume":"86","author":"Deppe","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1088\/0264-9381\/24\/8\/013","article-title":"Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach","volume":"24","author":"Capozziello","year":"2007","journal-title":"Class. Quant. Grav."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"4671","DOI":"10.1088\/0264-9381\/19\/18\/304","article-title":"General Gauss\u2013Bonnet brane cosmology","volume":"19","author":"Charmousis","year":"2002","journal-title":"Class. Quant. Grav."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/372\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:03:27Z","timestamp":1760173407000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,2]]},"references-count":55,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030372"],"URL":"https:\/\/doi.org\/10.3390\/sym12030372","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,3,2]]}}}