Ideas of E.Cartan and S.Lie in modern geometry: GGG-structures and differential equations. Lecture 2 (original) (raw)
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The Frenet Frame and Space Curves
2019
Essential to the study of space curves in Differential Geometry is the Frenet frame. In this thesis we generate the Frenet equations for the second, third, and fourth dimensions using the Gram-Schmidt process, which allows us to present the form of the Frenet equatins for n-dimensions. We highlight several key properties that arise from the Frenet equations, expound on the class of curves with constant curvature ratios, as well as characterize spherical curves up to the fourth dimension. Methods for generalizing properties and characteristics of curves in varying dimensions should be handled with care, since the structure of curves often differ in progressing dimensions.
It is well known that Minkowski space is the mathematical space setting and Einstein's theory of special relativity is most appropriate formulated. Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems in which usually by employing di¤erential equations or di¤erence equations. The Frenet-Serret formulas describe the kinematic properties of a particle which moves along a continuous, di¤erentiable curve in Euclidean space three-dimensional real space or the geometric properties of the curve itself in any case of any motion. The Frenet-Serret trihedron plays a key role in the di¤erential geometry of curves such that its shows ultimately leading to a more or less complete classi…cation of smooth curves in Euclidean space up to congruence. In this paper, we established mechanics Equations of Frenet-Serret frame on Minkowski space and we considered a relativistic for an electromagnetic …eld that it is moving under the in ‡uence of its own Frenet-Serret curvatures. Also, we obtained the mechanical equations of motion for several curvatures dependent actions of interest in physics.
Characterizations of Adjoint Curves According to Alternative Moving Frame
Fundamental Journal of Mathematics and Applications
In this paper, the adjoint curve is defined by using the alternative moving frame of a unit speed space curve in 3-dimensional Euclidean space. The relationships between Frenet vectors and alternative moving frame vectors of the curve are used to offer various characterizations. Besides, ruled surfaces are constructed with the curve and its adjoint curve, and their properties are examined. In the last section, there are examples of the curves and surfaces defined in the previous sections.
The Frenet Vector Fields and the Curvatures of the Natural Lift Curve
The Bulletin of Society for Mathematical Services and Standards, 2012
In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in 3/1 . Also, a similar calculation is made in 3/1 considering timelike or spacelike Darboux vector field.
Differential Geometry and Relativity Theories vol. 1
2017
In this book, we focus on some aspects of smooth manifolds, which appear of fundamental importance for the developments of differential geometry and its applications to Theoretical Physics, Special and General Relativity, Economics and Finance. In particular we touch basic topics, for instance definition of tangent vectors, change of coordinate system in the definition of tangent vectors, action of tangent vectors on coordinate systems, structure of tangent spaces, geometric interpretation of tangent vectors, canonical tangent vectors determined by local charts, tangent frames determined by local charts, change of local frames, tangent vectors and contravariant vectors, covariant vectors, the gradient of a real function, invariant scalars, tangent applications, local Jacobian matrices, basic properties of the tangent map, chain rule, diffeomorphisms and derivatives, transformation of tangent bases under derivatives, paths on a manifold, vector derivative of a path with respect to a re-parametrization, tangent derivative versus calculus derivative, vector derivative of a path in local coordinates, existence of a path with a given initial tangent vector and other topics.
Frenet frame with respect to conformable derivative
Filomat, 2019
Conformable fractional derivative is introduced by the authors Khalil at al in 2014. In this study, we investigate the frenet frame with respect to conformable fractional derivative. Curvature and torsion of a conformable curve are defined and the geometric interpretation of these two functions is studied. Also, fundamental theorem of curves is expressed for the conformable curves and an example of the curve corresponding to a fractional differential equation is given.
On Inclined Curves According to Parallel Transport Frame in E4
In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "\alpha: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds = 0" where k_1, k_2, K_3 are the principal curvature functions according to parallel transport frame of the curve and we give the similar characterizations such as "\alpha : I \subset R \rightarrow E^3 is a generalized helix \Leftrightarrow k_1 \int k_1ds + k_2 \int k_2ds = 0" where k_1, k_2 are the principal curvature functions according to Bishop frame of the curve \alpha. Moreover, we illustrate some examples and draw their figures with Mathematica Programme.