Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.

Section: Discussionmentioning

confidence: 99%

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

Nazra,

Abdel-Baky

2023

“…Recall that there is a typical relationship between singularity theory and submanifold theory (cf. [7,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). Note that when a straight line moves along a curve, it sweeps out a surface known as a ruled surface in Euclidean 3-space [15].…”

Section: Discussionmentioning

confidence: 99%

A Note on Incompressible Vector Fields

Turki

2023

In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

“…Furthermore, Li et al engaged in theoretical research and applied their findings to the fields of solitons theory and submanifolds theory, etc. thus contributing to the advancement of these related research areas [28][29][30][31][32][33][34][35][36]. Their work demonstrates a deep understanding of mathematical concepts and highlights the potential applications of such theories.…”

Section: Introductionmentioning

confidence: 89%

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

Hakami,

Siddiqi

2023

In this manifestation, we explain the geometrisation of η-Ricci–Yamabe soliton and gradient η-Ricci–Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short CV) is an η-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the η-Ricci–Yamabe soliton in Riemannian submersions with a φ(Q)-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory.