Biharmonic Curves in Warped Product Manifolds $I\times_{f}M^{n}\left( c\right) $

TL;DR

This paper investigates the geometric properties of biharmonic curves in warped product manifolds, providing a comprehensive analysis of their curvature characteristics and constructing explicit examples in specific cases.

Contribution

It introduces new theoretical results on biharmonic curves in warped product manifolds and classifies their behavior in various curvature scenarios.

Findings

Characterization of biharmonic curves in warped product manifolds

Analysis of curvature-related properties including slant cases

Explicit examples in warped products involving 2-spheres

Abstract

We explore the geometric properties of biharmonic curves in warped product manifolds of the form $I\times _{f}M^{n}(c)$, where $I$ is an open interval and $M^{n}(c)$ is a space of constant curvature. By establishing a main theorem, we analyze four distinct cases to reveal deeper curvature-related characteristics of these curves, including situations where they are slant. Finally, we construct two examples in $I\times _{f}S^{2}(1)$.