Framed Natural Mates of Framed Curves in Euclidean 3-Space

Abstract

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples.

Significance of Solitonic Fibers in Riemannian Submersions and Some Number Theoretic Applications

“…Moreover, the pursuit of interdisciplinary research holds great promise for producing valuable insights. In forthcoming research, we intend to explore additional applications of the primary findings presented in this paper to soliton theory, submanifold theory, and related areas, as examined in [1,2,21,[44][45][46][47][48][49][50][51][52][53][54][55][56], with the goal of achieving new results.…”

Semi-Conformally Flat Singly Warped Product Manifolds and Applications

“…There are sufficient conditions on the warping function to ensure that the base manifold is a quasi-Einstein or an Einstein manifold. We can find more motivations of our work from the following papers [44][45][46][47][48][49][50][51].…”

Contact CR-Warped Product Submanifold of a Sasakian Space Form with a Semi-Symmetric Metric Connection

“…Proof. For any Λ 1 , Λ 2 ∈ TΩ T , and Λ 3 ∈ TΩ ⊥ , we utilize equation (19), the Gauss formula, along with equation (2), to obtain the following result…”

“…Moreover, theoretical research and development on submanifold theory, soliton theory, and related topics were carried out by researchers like Li et al, as evidenced by a series of referenced papers [12][13][14][15][16][17][18][19][20][21][22][23][24]. The works of Li et al have made significant contributions to the advancement of these research areas, providing valuable insights and motivation for further exploration.…”