f L k -Harmonic Maps and f L k -Harmonic Morphisms
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fLk -Harmonic Maps and fLk -Harmonic Morphisms Mehran Aminian1 · Mehran Namjoo1 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 f Lk -energy functionals; and by deriving variations of these functionals, we define f Lk -harmonic maps between Riemannian manifolds. Hereafter, by using these definitions, we introduce f Lk -harmonic morphisms, and then we find a relation between f Lk -harmonic maps and f Lk -harmonic morphisms. Keywords Lk 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 generalize energy functionals and the notions of tension fields to introduce Lk -harmonic maps. Following it, we introduce f Lk -energy functionals and by computing the first variation of these functionals, we define f Lk -harmonic maps between two Riemannian manifolds. After that, we introduce f Lk -harmonic morphisms and then we find a relation between f Lk -harmonic morphisms and f Lk -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 Rn+1 , for c = 0, and the hyperbolic space Hn+1 , for c = −1, and the Euclidean sphere Sn+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
[email protected] Mehran Namjoo [email protected] 1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
M. Aminian, M. Namjoo
connection on the pullback bundle ϕ ∗ T R n+1 (c) by ∇. Let X, Y be vector fields on M. We have the following formula for the shape operator of M: ∇ X dϕ(Y ) = dϕ(∇X Y ) + SX, Y N,
dϕ(SX) = −∇ X N .
As it is known, the shape operator is a self-adjoint linear operator. Let k1 , . . . , kn be its eigenvalues which are called principal curvatures of M. Define s0 = 1 and sk = ki1 · · · kik . 1≤i1
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