Abstract
We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.
Highlights
An arc-length parametrized curve γ : I → M n from an open interval I ⊂ R to a Riemannian manifold of dimension n is called triharmonic if∇5T T + RM (∇3T T, T ) T − RM (∇2T T, ∇T T ) T = 0 where T is the unit tangent vector field of γ, ∇ denotes the Levi-Civita connection of M n and RM is the Riemannian curvature tensor of M n.Triharmonic curves represent the case r = 3 in a general theory of r-harmonic curves
The case r = 2, that is of biharmonic curves, is well studied and it is well known that if we denote by κ(s) = ∇T T the curvature of an arc-length parametrized curve γ : I → M n in a Riemannian manifold M n, if γ is proper biharmonic the curvature κ is constant
In the first part of the paper, we investigate the possibility of constructing triharmonic curves in a Riemannian manifold with nonconstant curvature and we obtain the following result
Summary
An arc-length parametrized curve γ : I → M n from an open interval I ⊂ R to a Riemannian manifold of dimension n is called triharmonic if. The case r = 2, that is of biharmonic curves, is well studied and it is well known (see, for example, [5]) that if we denote by κ(s) = ∇T T the curvature of an arc-length parametrized curve γ : I → M n in a Riemannian manifold M n, if γ is proper biharmonic the curvature κ is constant. We first prove that triharmonic curves with constant curvature in a Riemannian manifold of dimension 3 are Frenet helices (Corollary 4.2). The latter result enables us to tackle the classification problem of triharmonic curves with constant curvature in homogeneous 3-dimensional manifolds. It turns out that these triharmonic curves can be seen as geodesics of suitable Hopf cylinders (see Corollary 4.8)
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