University of Kashan Mathematics Interdisciplinary Research 2538-3639 11 3 2026 09 01 Generalized Recurrent and $\psi$-Recurrent Curvature on Mixed 3-Sasakian Manifolds 201 212 115556 10.22052/mir.2026.257679.1543 EN Fatemeh Asali ‎Department of Mathematics‎, University of Zanjan, ‎Zanjan‎, ‎I‎. ‎R‎. ‎Iran 0009-0005-7614-4101 Mohammad Bagher Kazemi Balgeshir ‎Department of Mathematics‎, University of Zanjan, ‎Zanjan‎, ‎I‎. ‎R‎. ‎Iran 0000-0002-6915-6754 Fatemeh Raei ‎Department of Mathematics Education‎, ‎Farhangian University,‎ ‎Tehran‎, ‎I. R. ‎Iran Journal Article 2025 09 28 ‎Considering the importance of mixed 3-Sasakian manifolds which admit Einstein metrics‎, ‎we introduce generalized mixed 3-$ \psi $-recurrent manifolds on mixed 3-Sasakian structures‎. ‎We prove that a non-flat generalized mixed 3-$\psi$-recurrent manifold is a generalized recurrent manifold with its horizontal vector fields‎. ‎Also‎, ‎we obtain a relation between associated 1-forms $\gamma_{i} $'s‎ and $\theta_{i} $'s for a generalized mixed 3-$ \psi$-recurrent manifold‎. ‎Moreover‎, ‎we give a necessary and sufficient condition for a mixed 3-Sasakian manifold to be a generalized mixed 3-$\psi$-recurrent‎. ‎Next‎, ‎we find the Riemannian curvature representation of a mixed 3-Sasakian manifold when that is a generalized mixed 3-$\psi$-recurrent‎. Mixed 3-structures‎ ‎Recurrent manifold‎ ‎Generalized 3-$ \psi $-recurrent manifold‎ ‎Einstein manifold https://mir.kashanu.ac.ir/article_115556_8b0e08d6a104e3a3ee0d6f1d7ad3307b.pdf