One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes

Abstract

E. Study map is one of the most basic and powerful mathematical tools to study lines in line geometry, it has symmetry property. In this paper, based on the E. Study map, clear expressions were developed for the differential properties of one-parameter Lorentzian dual spherical movements that are coordinate systems independent. This eliminates the requirement of demanding coordinates transformations necessary in the determination of the canonical systems. With the proposed technique, new proofs for Euler–Savary, and Disteli’s formulae were derived.

A Study of Conformal $$\eta$$-Einstein Solitons on Trans-Sasakian 3-Manifold

“…On using Eq. ( 70) with (37) we have Therefore we can state the following: Theorem 13 Let (M, g) be a trans-Sasakian 3-manifold admitting a conformal -Einstein soliton (g, , , ) . If the manifold satisfies the curvature condition W 2 ( , V 1 ) ⋅ S = 0 then either the manifold becomes an Einstein manifold or it is a manifold of constant scalar curvature r =…”

One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode

“…In this section, we give a short synopsis of the dual numbers theory, and dual Lorentzian vectors [11][12][13][14][15]. If a and a * are real numbers, the term a = a + εa * is named a dual number.…”

“…The center and radius of this circle can be specified by the Euler-Savary formula, if the place of the point is specified in the movable plane. In different types of geometry, the Euler-Savary formula had been generalized for a line trajectory, i.e., the construction of the Disteli formulae [4][5][6][7][11][12][13] . Therefore, we now shall look to the Euler-Savary and Disteli formulae for the timelike axodes by utilizing the equipment just acquired above.…”

“…In the Minkowski 3-space E 3 1 , Lorentzian metric can have one of three Lorentzian causal characters; it can be positive, negative, or zero, while the metric in the Euclidean 3-space E 3 can only be positive definite. Therefore, the kinematic and geometric clarifications can be further meaningful in E 3 1 [11][12][13]. This study is as follows.…”

Conformal η-Ricci Solitons on Riemannian Submersions under Canonical Variation

“…Riemannian submersions have been intensively investigated not only in mathematics, but also in theoretical physics due to its usefulness in Kaluza Klein theory, super gravity, Yang-Mills theory, relativity, and super-string theories (see [9][10][11][12][13]). Singularity theory and submanifold theory are also crucially related to this subject and will be helpful for future research (for more details see [14][15][16][17]). The majority of Riemannian submersion investigations may be found in books [18,19].…”