{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:17:59Z","timestamp":1760059079212,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,18]],"date-time":"2025-05-18T00:00:00Z","timestamp":1747526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Serbian Ministry of Science, Technological Development and Innovation","award":["451-03-137\/2025-03\/200122"],"award-info":[{"award-number":["451-03-137\/2025-03\/200122"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce null Cartan normal helices in Minkowski space E13. We obtain explicit expressions for their torsions by considering the cases when the C-constant vector field is orthogonal to their axis or not orthogonal to it. We find that the tangent vector field of a null Cartan normal helix satisfies the third-order linear homogeneous differential equation and obtain its general solution in a special case. We prove that null Cartan helices are the only normal helices having two axes and, in a particular case, three axes. Finally, we provide the necessary and sufficient conditions for null Cartan normal helices lying on a timelike surface to be isophotic curves, silhouettes, normal isophotic curves and normal silhouettes with respect to the same axis and provide some examples.<\/jats:p>","DOI":"10.3390\/axioms14050379","type":"journal-article","created":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T06:31:29Z","timestamp":1747636289000},"page":"379","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Null Cartan Normal Helices in Minkowski 3-Space"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3124-6308","authenticated-orcid":false,"given":"Emilija","family":"Ne\u0161ovi\u0107","sequence":"first","affiliation":[{"name":"Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1158","DOI":"10.55730\/1300-0098.3418","article-title":"A generalization of the notion of helix","volume":"47","author":"Lucas","year":"2023","journal-title":"Turk. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"650","DOI":"10.3906\/mat-1410-4","article-title":"On isophote curves and their characterizations","volume":"39","author":"Dogan","year":"2015","journal-title":"Turk. J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7583","DOI":"10.1002\/mma.5221","article-title":"On k\u2013type null Cartan slant helices in Minkowski 3-space","volume":"41","year":"2018","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2450142","DOI":"10.1142\/S0219887824501421","article-title":"Null Cartan geodesic isophote curves in Minkowski 3-space","volume":"21","author":"Li","year":"2024","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"10520","DOI":"10.1002\/mma.10137","article-title":"On null Cartan normal isophotic and normal silhouette curves on a timelike surface in Minkowski 3-space","volume":"47","year":"2024","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_6","unstructured":"O\u2019Neill, B. (1983). Semi-Riemannian Geometry with Applications to Relativity, Academic Press."},{"key":"ref_7","first-page":"229","article-title":"Null curves in a Minkowski space-time","volume":"20","author":"Bonnor","year":"1969","journal-title":"Tensor"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Duggal, K.L., and Jin, D.H. (2007). Null Curves and Hypersurfaces of Semi\u2013Riemannian Manifolds, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/6449"},{"key":"ref_9","unstructured":"Walrave, J. (1995). Curves and Surfaces in Minkowski Space. [Ph.D. Thesis, Katholieke Universiteit Leuven]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/379\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:34:44Z","timestamp":1760031284000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/379"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,18]]},"references-count":9,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050379"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050379","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,18]]}}}