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

TL;DR

The study investigates biharmonic curves in warped product manifolds of the form $I\times_f M^{n}(c)$ by deriving the biharmonic equation for unit-speed curves using the warped-product geometry and Frenet apparatus. The authors prove a main theorem characterizing biharmonicity in terms of a primary warping-condition $\left.\left[\frac{f''}{f}-\left(\frac{f'}{f}\right)^2+\frac{c}{f^{2}}\right]\right|_{\gamma}$, the structural angle via $\eta(E_{2})$, and the span of the structural vector $\partial_t$, together with the first $m=\min\{r,4\}$ curvature equations along the curve. Four cases are analyzed to reveal when biharmonic curves are geodesic, circles, or helices, and how the warping function $f$ interacts with curvature data. The paper also provides two explicit examples in $I\times_f S^{2}(1)$ with $f(t)=\sin t$ (a biharmonic circle and a biharmonic helix), illustrating the practical application of the theory and validating the derived corollaries. This framework advances the classification of biharmonic curves in warped product spaces and invites extension to broader warped geometries and higher dimensions.

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 $I\times _{f}S^{2}(1)$.

Theorems & Definitions (23)

• Remark 1
• Proposition 1
• proof
• Remark 2
• Lemma 1
• Theorem 1
• proof
• Corollary 1
• Remark 3
• Lemma 2