Acta Mathematica Vietnamica https://doi.org/10.1007/s40306-020-00392-0 fL k -Harmonic Maps and fL k -Harmonic Morphisms Mehran Aminian 1 · Mehran Namjoo 1 Received: 29 December 2018 / Revised: 18 May 2020 / Accepted: 23 June 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020 Abstract In this paper, we introduce fL k -energy functionals; and by deriving variations of these functionals, we define fL k -harmonic maps between Riemannian manifolds. Hereafter, by using these definitions, we introduce fL k -harmonic morphisms, and then we find a relation between fL k -harmonic maps and fL k -harmonic morphisms. Keywords L k operator · Energy functional · Harmonic map Mathematics Subject Classification (2010) 58E20 · 53C43 1 Introduction and Preliminaries Harmonic maps are critical points of energy functionals; equivalently, these maps are solutions of PDE systems when tension fields are zero [7, 10]. In paper [2], the authors gen- eralize energy functionals and the notions of tension fields to introduce L k -harmonic maps. Following it, we introduce fL k -energy functionals and by computing the first variation of these functionals, we define fL k -harmonic maps between two Riemannian manifolds. After that, we introduce fL k -harmonic morphisms and then we find a relation between fL k -harmonic morphisms and fL k -harmonic maps. In the paper, we used techniques of [12] to get the results. We recall the prerequisites from [1, 4–6, 11, 13]. Let R n+1 (c) be the simply connected Riemannian space form of constant sectional curvature c which is the Euclidean space R n+1 , for c = 0, and the hyperbolic space H n+1 , for c =−1, and the Euclidean sphere S n+1 , for c =+1. Let ϕ : M n → R n+1 (c) be a connected oriented hypersurface isometrically immersed into R n+1 (c) with N as a unit normal vector field, ∇ and ∇ the Levi-Civita connections on M and R n+1 (c), respectively. For simplicity, we also denote the induced  Mehran Aminian mehran.aminian@vru.ac.ir Mehran Namjoo namjoo@vru.ac.ir 1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran