Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 5 2 2024 03 01 A generalization of normal almost contact manifolds 233 241 208727 10.30504/jims.2024.435257.1158 EN F. Malek Faculty of‎ ‎Mathematics, K‎. ‎N‎. ‎Toosi University of Technology, Tehran‎, ‎Iran. 0000-0001-6504-4184 A. Ghoujaei Bavil Oliaei Department of Mathematics, Payam Noor University Tehran, Iran. Journal Article 2024 01 11 In this article‎, ‎a new definition‎, ‎called $ \varphi $-normal‎, ‎is introduced‎, ‎which is a generalization of normal condition on almost contact manifolds‎. ‎Then some examples of $ \varphi $-normal almost contact manifolds that are not normal are presented‎, ‎and a sufficient and necessary condition for equivalence of these two definitions in 3-dimensional almost contact manifolds is provided‎. ‎In the end‎, ‎it is proven that a $ \varphi$-normal contact metric manifold is a Sasakian manifold‎. ‎Sasakian manifolds‎ ‎Nijenhuis torsion‎ ‎normal almost contact manifolds https://jims.ims.ir/article_208727_40c0d8b7bbf8d2372a85b4695b1cd0ac.pdf