Polyharmonic curves in semi-Riemannian manifolds

TL;DR

This paper investigates the existence and classification of polyharmonic Frenet curves in semi-Riemannian manifolds, especially in space forms, ruled Lorentzian surfaces, and warped products, revealing conditions for their existence.

Contribution

It provides new existence, non-existence, and classification results for polyharmonic Frenet curves in various semi-Riemannian ambient spaces.

Findings

Existence conditions for polyharmonic Frenet curves in space forms.

Non-existence results in certain semi-Riemannian manifolds.

Classification of polyharmonic curves in specific geometric settings.

Abstract

Let $(M^m_t,g)$ be a semi-Riemannian manifold of dimension $m$ with a non-degenerate metric of \textit{index} $t$, $m\geq 2$, $1 \leq t \leq m-1$. The main aim of this paper is to investigate the existence of Frenet curves in $(M^m_t,g)$ which are polyharmonic of order $r$, shortly, $r$-harmonic. We shall focus primarily on the cases that the ambient space is a semi-Riemannian space form $N^m_t(c)$ of sectional curvature $c$, a ruled Lorentzian surface or a suitable, possibly warped, product space. We shall obtain existence, non-existence and classification results.