{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:09Z","timestamp":1760145069123,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Project of Science and Technology of Heilongjiang Provincial Education Department","award":["1453ZD029","1453ZD019","ZYQN2019071"],"award-info":[{"award-number":["1453ZD029","1453ZD019","ZYQN2019071"]}]},{"name":"Reform and Development Foundation for Local Colleges and Universities of the Central Government","award":["1453ZD029","1453ZD019","ZYQN2019071"],"award-info":[{"award-number":["1453ZD029","1453ZD019","ZYQN2019071"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss\u2013Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group.<\/jats:p>","DOI":"10.3390\/sym16060762","type":"journal-article","created":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T07:33:20Z","timestamp":1718696000000},"page":"762","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Gauss\u2013Bonnet Theorem Related to the Semi-Symmetric Metric Connection in the Heisenberg Group"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0410-5980","authenticated-orcid":false,"given":"Haiming","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics Science, Mudanjiang Normal University, Mudanjiang 157011, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Song","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Mathematics Science, Mudanjiang Normal University, Mudanjiang 157011, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,18]]},"reference":[{"key":"ref_1","first-page":"640","article-title":"An introduction to Heisenberg groups in analysis and geometry","volume":"50","author":"Semmes","year":"2003","journal-title":"Not. 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