Submanifold Differential Operators in $\Cal D$-Module Theory II: Generalized Weierstrass and Frenet-Serret Relations as Dirac Equations

A primer on the differential geometry of quaternionic curves

This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes benefit of the quaternionic structure and the results are much simpler than the present formulations of quaternionic curves.

On the differential geometry of curves in Minkowski space

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions of a very known theorem of the differential geometry of curves in tridimensional Euclidean space. We discuss the general solution for torsionless paths in Minkowki space. We then apply the four-dimensional Serret-Frenet equations to describe the motion of a charged test particle…

Closed Constant Curvature Space Curves

We use ODEs and symmetry arguments to construct closed constant curvature space curves, first on cylinders, next on tori, at last with the Frenet-Serret equations.

非平坦三维空间形式$M_{q}^{3}(c)$中非零曲线的电磁场几何与麦克斯韦方程札记

Notes on the Geometry of Electromagnetic Fields and Maxwell's Equations along a non-null curves in non flat-3D space forms $M_{q}^{3}(c)$

该研究在非平坦三维空间形式$M_{q}^{3}(c)$中定义正交标架下的方向导数、推广Serret-Frenet关系，给出粒子弯曲弹性能函数及单位向量场能量的几何解释，并求解该空间中的麦克斯韦方程。

Some Characterizations of Special Curves in the Euclidean Space $\mathrm{E}^4$

In this work, first, we express some characterizations of helices and ccr curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret invariants of Bertrand curve of a helix are presented. Moreover, in the same space, some new characterizations of involute of a helix are presented.

Submanifold Differential Operators in $\Cal D$-Module Theory I : Schrödinger Operators

For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated in such a scheme , which are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case. These Dirac operators are concerned well in the diffe…

Geometry of Curves in $\mathbb R^n$, Singular Value Decomposition, and Hankel Determinants

Let $γ: I \rightarrow \mathbb R^n$ be a parametric curve of class $C^{n+1}$, regular of order $n$. The Frenet-Serret apparatus of $γ$ at $γ(t)$ consists of a frame $e_1(t), \dots , e_n(t)$ and generalized curvature values $κ_1(t), \dots, κ_{n-1}(t)$. Associated with each point of $γ$ there are also local singular vectors $u_1(t), \dots, u_n(t)$ and local singular values $σ_1(t), \dots, σ_{n}(t)$. This local information is obtained by considering a limit, as $ε$ goes to zero, of covariance matrices defined along $γ$ within an $ε$-ball centered at $γ(t)$. We prove that for each $t\in I$, the Fre…

Differential galois theory and mechanics

The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of algebraic partial differential (PD) equations and the corresponding finitely generated differential extensions, in order to be able to add the word differential in front of any classical statement. The achievement of the Pi…