{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:15:11Z","timestamp":1760238911520,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"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 derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami\u2013Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami\u2013Poincare disk model, known as M\u00f6bius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.<\/jats:p>","DOI":"10.3390\/sym12091525","type":"journal-article","created":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T10:30:12Z","timestamp":1600252212000},"page":"1525","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Differential Geometry and Binary Operations"],"prefix":"10.3390","volume":"12","author":[{"given":"Nikita E.","family":"Barabanov","sequence":"first","affiliation":[{"name":"Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2882-1663","authenticated-orcid":false,"given":"Abraham A.","family":"Ungar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.jmaa.2016.11.039","article-title":"Finitely generated gyrovector subspaces and orthogonal gyrodecomposition in the M\u00f6bius gyrovector space","volume":"449","author":"Abe","year":"2017","journal-title":"J. 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